CCOXETER-BICATALAN COMBINATORICS
EMILY BARNARD AND NATHAN READING
Abstract.
We pose counting problems related to the various settings forCoxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian).Each problem is to count “twin” pairs of objects from a corresponding prob-lem in Coxeter-Catalan combinatorics. We show that the problems all havethe same answer, and, for a given finite Coxeter group W , we call the com-mon solution to these problems the W -biCatalan number. We compute the W -biCatalan number for all W and take the first steps in the study of Coxeter-biCatalan combinatorics. Contents
1. Introduction 22. BiCatalan objects 52.1. Antichains in the doubled root poset and twin nonnesting partitions 52.2. BiCambrian fans 82.3. The biCambrian congruence, twin sortable elements, and bisortableelements 122.4. Twin clusters and bicluster fans 142.5. Twin noncrossing partitions 153. Bipartite c -bisortable elements and alternating arc diagrams 173.1. Pattern avoidance 173.2. Noncrossing arc diagrams 183.3. Alternating arc diagrams 193.4. Counting alternating arc diagrams 203.5. Enumerating bipartite c -bisortable elements in type B 253.6. Simpliciality of the bipartite biCambrian fan in types A and B 284. Double-positive Catalan numbers and biCatalan numbers 314.1. Double-positivity 324.2. Counting twin nonnesting partitions 334.3. Canonical join representations and lattice congruences 354.4. Canonical join representations of c -bisortable elements 384.5. Counting bipartite c -bisortable elements 404.6. BiCatalan and Catalan formulas 434.7. The double-positive Catalan numbers 474.8. The Type D biCatalan number 514.9. Type-D biNarayana numbers 53Acknowledgments 53References 53 Emily Barnard was supported in part by NSF grants DMS-0943855, DMS-1101568, and DMS-1500949. Nathan Reading was supported in part by NSF grants DMS-1101568 and DMS-1500949. a r X i v : . [ m a t h . C O ] J un EMILY BARNARD AND NATHAN READING Introduction
This paper considers enumeration problems closely related to Coxeter-Catalancombinatorics. (For background on Coxeter-Catalan combinatorics, see for example[5, 18]). Each enumeration problem can be thought of as counting pairs of “twin”Coxeter-Catalan objects—twin sortable elements or twin nonnesting partitions, etc.Many of the terms used in this introductory section are new to this paper and willbe explained in Section 2.In the setting of sortable elements and Cambrian lattices/fans, the enumerationproblem is to count the following families of objects: • maximal cones in the bipartite biCambrian fan (the common refinement oftwo bipartite Cambrian fans); • pairs of twin c -sortable elements for bipartite c ; • classes in the bipartite biCambrian congruence (the meet of two bipartiteCambrian congruences); • elements of the bipartite biCambrian lattice; • c -bisortable elements for bipartite c .In type A, c -bisortable elements for bipartite c are in bijection with permutationsavoiding a set of four bivincular patterns in the sense of [8, Section 2] and withalternating arc diagrams, as will be explained in Sections 3.1–3.3. In type B, similarbijections exist with certain signed permutations and with centrally symmetricalternating arc diagrams, as described in Section 3.5.In the setting of nonnesting partitions (antichains in the root poset), the enu-meration problem is to count two families of objects: • antichains in the doubled root poset; • pairs of twin nonnesting partitions.In the setting of clusters of almost positive roots (in the sense of [20]), theproblem is to count two families of objects: • maximal cones in the bicluster fan (the common refinement of the clus-ter fan, in the original bipartite sense of Fomin and Zelevinsky, and itsantipodal opposite); • pairs of twin clusters, again in the bipartite sense.In the setting of noncrossing partitions, the problem is to count the followingfamilies of objects: • pairs of twin bipartite c -noncrossing partitions; • pairs of twin bipartite ( c, c − )-noncrossing partitions.The main result of this paper is the following. Theorem 1.1.
For each finite Coxeter group/root system, all of the enumerationproblems posed above have the same answer.
In all of the settings above except the nonnesting setting, the objects describedabove can be defined for arbitrary choices of a Coxeter element. However, theenumerations depend on the choice of Coxeter element, and we emphasize thatTheorem 1.1 is an assertion about the enumeration in the case where the Coxeterelement is chosen to be bipartite. See Section 2.2 for the definition of Coxeterelements and bipartite Coxeter elements.The enumeration problems in the nonnesting setting require a crystallographicroot system, but outside of the nonnesting setting, Theorem 1.1 still holds for
OXETER-BICATALAN COMBINATORICS 3 noncrystallographic types. For each noncrystallographic type except H , one candefine a root poset, and Theorem 1.1 holds; see Remark 2.1.We will see in Section 2 that within each group of bullet points above, thevarious enumeration problems have the same answer essentially by definition. Usingknown uniform correspondences from the usual Coxeter-Catalan combinatorics, itis straightforward to give (in Theorems 2.19 and 2.22) uniform bijections connectingthe Cambrian/sortable setting to the noncrossing and cluster settings. The difficultpart of the main result is the following theorem which connects the nonnestingsetting to the other settings. Theorem 1.2.
For crystallographic W , c -bisortable elements for bipartite c are inbijection with antichains in the doubled root poset. More specifically, we have the following refined version of Theorem 1.2.
Theorem 1.3.
For crystallographic W and for any k , the number of bipartite c -bisortable elements with k descents equals the number of k -element antichains inthe doubled root poset. Our proof of Theorems 1.2 and 1.3 in Section 4 would be uniform if a uniformproof were known connecting the nonnesting setting to the other settings of theusual Coxeter-Catalan combinatorics. Indeed, the opposite is true: A well-behaveduniform bijection proving Theorem 1.2 or Theorem 1.3 would imply a uniformproof of the analogous Coxeter-Catalan statement. (See Remark 4.30 for details.)However, the proofs of these theorems are far from a trivial recasting of Coxeter-biCatalan combinatorics in terms of Coxeter-Catalan combinatorics. Instead, itrequires a count of antichains in the doubled root poset indirectly in terms of theCoxeter-Catalan numbers and a nontrivial proof that the same formula holds forbipartite c -bisortable elements. The formula uses a notion of “double-positive”Catalan and Narayana numbers, which already appeared in [3] as the local h -polynomials of the positive cluster complex. (See Remark 4.7 and Theorem 4.52.)We propose the terms W -biCatalan number and W -biNarayana number and the symbols biCat( W ) and biNar k ( W ) for the numbers appearing in Theo-rems 1.1 and 1.3. Theorem 1.4.
The W -biCatalan numbers for irreducible finite Coxeter groups are: W A n B n D n E E E F H H I ( m )biCat( W ) (cid:0) nn (cid:1) n − · n − − (cid:0) n − n − (cid:1) m . The type-A and type-B cases of Theorem 1.4 are proved, in the nonnestingsetting, in Section 2.1 by recasting the antichain count as a count of lattice paths.The same cases can also be established in the setting of c -bisortable elements byrecasting the problem in terms of alternating arc diagrams. Although the latterapproach is more difficult, we carry out the type-A and type-B enumeration bythe latter approach in Section 3, because the combinatorial models for bipartite c -bisortable elements in types A and B are of independent interest, and becausethe enumeration of alternating arc diagrams provides the crucial insight whichleads to the recursive proof of Theorem 1.1. (See Remark 3.13.) The type-D caseof Theorem 1.4 is much more difficult, and involves solving the type-D case ofthe recursion used in the proof of Theorem 1.2. The formula in type D was firstguessed using the package GFUN [42]. The enumerations in the exceptional typeswere obtained using Stembridge’s posets and coxeter/weyl packages [46].
EMILY BARNARD AND NATHAN READING
We also obtain formulas for the W -biNarayana numbers outside of type D. InSection 4, we write biCat( W, q ) for the polynomial in the second column in thetables below.
Theorem 1.5.
The biNarayana numbers of irreducible finite Coxeter groups, exceptin type D, are given by the following generating functions. W (cid:80) nk =0 biNar k ( W ) q k A n (cid:80) nk =0 (cid:0) nk (cid:1) q k B n (cid:80) nk =0 (cid:0) n k (cid:1) q k E q + 415 q + 736 q + 415 q + 66 q + q E q + 1139 q + 3177 q + 3177 q + 1139 q + 119 q + q E q + 3226 q + 13210 q + 20728 q + 13210 q + 3226 q + 232 q + q F q + 106 q + 44 q + q G q + q H q + 27 q + q H q + 316 q + 116 q + q I ( m ) 1 + (2 m − q + q . Generating functions for biNarayana numbers for some type-D Coxeter groupsare shown here. At present we have no conjectured formula for the D n -biNarayananumbers. See Section 4.9 for a modest conjecture. D q + 42 q + 20 q + q D q + 136 q + 136 q + 35 q + q D q + 343 q + 600 q + 343 q + 54 q + q D q + 731 q + 2011 q + 2011 q + 731 q + 77 q + q D q + 1384 q + 5556 q + 8638 q + 5556 q + 1384 q + 104 q + q D q + 2402 q + 13314 q + 29868 q +29868 q + 13314 q + 2402 q + 135 q + q D q + 3901 q + 28624 q + 87874 q + 126336 q +87874 q + 28624 q + 3901 q + 170 q + q Naturally, one would like a uniform formula for the W -biCatalan number, but wehave not found one. A tantalizing near-miss is the non-formula (cid:81) ni =1 h + e i − e i , where h is the Coxeter number and the e i are the exponents. This expression captures the W -biCatalan numbers for W of types A n , B n , H , and I ( m )—the “coincidentaltypes” of [49]—but fails to even be an integer in some other types. In every case,the expression is a surprisingly good estimate of the W -biCatalan number.Section 2 is devoted to filling in definitions and details for the discussion aboveand proving the easy parts of Theorem 1.1. In Section 3, we explain why, in typeA, the bipartite bisortable elements are in bijection with alternating arc diagramsand carry out the enumeration of alternating arc diagrams. We carry out a similarenumeration in type B, in terms of centrally symmetric alternating arc diagrams.We conjecture that the bipartite biCambrian fan is simplicial (and thus that its dualpolytope is simple), and prove the conjecture in types A and B. In Section 4, we OXETER-BICATALAN COMBINATORICS 5 discuss double-positive Coxeter-Catalan numbers and establish a formula countingantichains in the doubled root poset in terms of double-positive Coxeter-Catalannumbers. We then show that bipartite c -bisortable elements satisfy the same recur-sion, thus proving Theorem 1.3 and completing the proof of Theorem 1.1. Finally,we establish some additional formulas involving double-positive Coxeter-Catalannumbers, Coxeter-Catalan numbers, and Coxeter-biCatalan numbers and use themto prove the formula for biCat( D n ) and thus complete the proof of Theorem 1.4.2. BiCatalan objects
In this section, we fill in the definitions and details behind the enumerationproblems discussed in the introduction. An exposition in full detail would requirereviewing Coxeter-Catalan combinatorics in full detail, so we leave some details tothe references.2.1.
Antichains in the doubled root poset and twin nonnesting partitions.
The root poset of a finite crystallographic root system Φ is the set of positive rootsin Φ, partially ordered by setting α ≤ β if and only if β − α is in the nonnegativespan of the simple roots. Recall that the dual of a poset ( X, ≤ ) is the poset ( X, ≥ ).That is, the dual has the same ground set, with x ≤ y in the dual poset if andonly if x ≥ y in the original poset. The doubled root poset consists of the rootposet, together with a disjoint copy of the dual poset, identified on the simple roots.Figure 1 shows some doubled root posets.The antichain counts in types A and B are easy and known, in the guise of latticepath enumeration. Antichains in the doubled root poset of type A n are in an easybijection with lattice paths from (0 ,
0) to ( n, n ) with steps (1 ,
0) and (0 , , , k right turns, we need only specifywhere the right turns are. This means choosing 0 ≤ x < · · · < x k ≤ n − ≤ y < · · · < y k ≤ n and placing right turns at ( x , y ) , . . . , ( x k , y k ). Thus, as iswell-known, there are (cid:0) nk (cid:1) paths with k right turns.Antichains in the doubled root poset of type B n are similarly in bijection withlattice paths from ( − n + 1 , − n + 1) to (2 n − , n −
1) with steps (2 ,
0) and (0 , y = − x . The k -element antichains correspond to paths with either 2 k right turns, ( k of which areto the left of the line y = − x ) or 2 k − k − y = − x and one of which is on the line y = − x ). Each path is uniquely determinedby its first 2 n − y = − x . Thus, thepaths map bijectively to words of length 2 n − N and E (for Northsteps (0 ,
2) and East steps (2 , E to the end of each word,the k -element antichains correspond to the words having exactly k positions wherean E appears immediately after an N . (The number of right turns in the path isodd if and only if one of these is position 2 n .) The 2 n -letter words ending in E andhaving exactly k instances of an E following an N are in bijection with 2 k -elementsubsets of { , . . . , n } . (Given such a word, take the set of positions where the letterchanges, with the convention that an N in the first position is a change but an E in the first position is not. So, for example, EN N EEE gives the subset { , } and EMILY BARNARD AND NATHAN READING A B D D F Figure 1.
Some doubled root posets
N EEEN E gives { , , , } .) We see that there are (cid:0) n k (cid:1) k -element antichains, and2 n − total antichains, in the doubled root poset of type B n . Remark 2.1.
It is not clear in general how one should define a “root poset” fora noncrystallographic root system. See [5, Section 5.4.1] for a discussion. In type I ( m ), there is an obvious way to define an unlabeled poset generalizing the rootposets of types A , B , and G . We say “unlabeled” here because it is obvious howthe poset should look but not obvious how the poset elements should correspond toroots. There is also a type- H root poset suggested in [5, Section 5.4.1]. For thesechoices of root posets, one can verify that Theorem 1.1 holds in these types as well. Remark 2.2.
The doubled root poset, and similar posets, were probably firstconsidered by Proctor (see [47, Remark 4.8(a)]) and then by Stembridge, as a toolfor counting reduced expressions for certain elements of finite Coxeter groups. Inthe simply-laced types (A, D, and E), the doubled root poset corresponds to the smashed Cayley order defined by Stembridge in [47, Section 4]. In the non-simply laced types, the smashed Cayley order is disconnected and is a strictlyweaker partial order than the doubled root poset. Stembridge [47, Theorem 4.6]shows that the component whose elements are short roots is a distributive lattice.Thus in particular the doubled root posets of types A, D, and E are distributivelattices. One can easily check distributivity in the remaining crystallographic types
OXETER-BICATALAN COMBINATORICS 7 A B D H F E E E Figure 2.
Some posets of join-irreducibles of doubled root posetsB, F, and G (and in fact in types H and I ( m )). By the Fundamental Theorem ofDistributive Lattices [45, Theorem 3.4.1], the doubled root poset is isomorphic tothe poset of order ideals in its subposet of join-irreducible elements. These posets ofjoin-irreducible elements are shown in Figure 2 for several types. An explicit root-theoretic description of the poset of join-irreducible elements in the simply-lacedtypes also appears in [47, Theorem 4.6].The support of a root β is the set of simple roots appearing with nonzerocoefficient in the expansion of β in the basis of simple roots. The support of a setof roots is the union of the supports of the roots in the set. We write ∆ for thesimple roots and, given a set A of roots, we write A ◦ for the set of non-simple rootsin A . If A and A are nonnesting partitions (i.e. antichains in the root poset), then( A , A ) is a pair of twin nonnesting partitions if and only if A ∩ ∆ = A ∩ ∆,and supp( A ◦ ) ∩ supp( A ◦ ) = ∅ .Given an antichain A in the doubled root poset, define top( A ) to be the intersec-tion of A with the root poset that forms the top of the doubled root poset. Definebottom( A ) to be the intersection of A with the dual root poset that forms thebottom of the doubled root poset. Both top( A ) and bottom( A ) are sets of positiveroots. The following proposition is an immediate consequence of the observationthat a root β in the top part of the doubled root poset is related to a root γ inthe bottom part of the doubled root poset if and only if the supports of β and γ overlap. EMILY BARNARD AND NATHAN READING
Proposition 2.3.
The map A (cid:55)→ (top( A ) , bottom( A )) is a bijection from an-tichains in the doubled root poset to pairs of twin nonnesting partitions. We pause to observe that the first biNarayana number (the number of elementsof the doubled root poset) is the number of roots minus the rank of W . Proposition 2.4. If W is an irreducible finite Coxeter group with Coxeter number h and rank n , then biNar ( W ) = n ( h − . BiCambrian fans.
The Cambrian fan is a complete simplicial fan whosemaximal faces are naturally in bijection [34, 39] with seeds in an associated clusteralgebra of finite type and with noncrossing partitions. Furthermore, the Cambrianfan is the normal fan [24, 25] to a simple polytope called the generalized associa-hedron [10, 20], which encodes much of the combinatorics of the associated clusteralgebra. More directly, the Cambrian fan is the g -vector fan of the cluster algebra.(This was conjectured, and proved in a special case, in [39, Section 10] and thenproved in general in [50].)The defining data of a Cambrian fan is a finite Coxeter group W and a Coxeterelement c of W . We emphasize that the results discussed in Section 1 concern aspecial “bipartite” choice of c , as explained below, but for now we proceed witha discussion for general c . A Coxeter element is the product of a permutationof the simple generators of W and may be specified by an orientation of the Cox-eter diagram. Given a choice of W , we will assume the usual representation of W as a reflection group acting with trivial fixed subspace. The collection of reflect-ing hyperplanes in this representation is the Coxeter arrangement of W . Thehyperplanes in the Coxeter arrangement cut space into cones, which constitute afan called the Coxeter fan F ( W ). The maximal cones of the Coxeter fan are inbijection with the elements of W . The Cambrian fan Camb ( W, c ) is the coarsen-ing of the Coxeter fan obtained by gluing together maximal cones according to anequivalence relation on W called the c -Cambrian congruence. Further details onthe c -Cambrian congruence appear in Section 2.3. For fixed W , all choices of c givedistinct but combinatorially isomorphic Cambrian fans.For each Coxeter element c , the inverse element c − is also a Coxeter element,corresponding to the opposite orientation of the diagram. We define the biCam-brian fan biCamb ( W, c ) to be the coarsest common refinement of the Cambrianfans
Camb ( W, c ) and
Camb ( W, c − ). Since Camb ( W, c ) and
Camb ( W, c − )are coarsenings of F ( W ), so is biCamb ( W, c ). Naturally, biCamb ( W, c − ) = biCamb ( W, c ). Example 2.5.
To illustrate the definition, take W of type B with simple genera-tors s and s . Figure 3 shows, from left to right, the s s -Cambrian fan, the s s -Cambrian fan, and the s s -biCambrian fan. Observe that the s s -biCambrian fancoincides with the B Coxeter fan. In general, when W is rank 2, the c -biCambrianfan for any choice of Coxeter element c is equal to the Coxeter fan F ( W ). Example 2.6.
For W of type A , there are two non-isomorphic c -biCambrian fans,shown in Figures 4 and 5 respectively. Each figure can be understood as follows:Intersecting the c -biCambrian fan with a unit sphere about the origin, we obtaina decomposition of the sphere into spherical convex polygons. The picture showsa stereographic projection of this polygonal decomposition to the plane. In eachcase, the walls of one Cambrian fan are shown in red and the walls of the opposite OXETER-BICATALAN COMBINATORICS 9
Figure 3.
Cambrian fans and the biCambrian fan in type B Cambrian fan are shown in blue. Walls that are in both Cambrian fans are showndashed red and blue.
Remark 2.7.
We observe that in Examples 2.5 and 2.6 that the common wallsof
Camb ( W, c ) and
Camb ( W, c − ) are exactly the reflecting hyperplanes for thesimple generators of W . This fact true in general, and the simplest proof involves shards . We will not define shards here, but definitions and results can be found,for example, in [36]. Assuming for a moment that background, we sketch a proof.First, recast [40, Theorem 8.3] as the statement that the c -Cambrian congruenceremoves all but one shard from each reflecting hyperplane of W . As explained inthe argument for [35, Proposition 1.3] (located in [35, Section 3] just after the proofof [35, Theorem 1.1]), the antipodal map sends the shard that is not removed bythe c -Cambrian congruence to the shard that is not removed by the c − -Cambriancongruence. The only shards that are fixed by the antipodal map are shards thatconsist of an entire reflecting hyperplane, and [36, Lemma 3.11] says that these areexactly the reflecting hyperplanes for the simple generators.The c − -Cambrian fan Camb ( W, c − ) coincides with − Camb ( W, c ), the imageof the c -Cambrian fan under the antipodal map. This is an immediate corollary of[35, Proposition 1.3], which is a statement about the c -Cambrian congruence. Seealso [41, Remark 3.27]. Thus we have the following proposition which amounts toan alternate definition of the biCambrian fan. Proposition 2.8.
The biCambrian fan biCamb ( W, c ) is the coarsest commonrefinement of Camb ( W, c ) and − Camb ( W, c ) . Since
Camb ( W, c ) and
Camb ( W, c − ) are the normal fans of two generalizedassociahedra, a standard fact (see [51, Proposition 7.12]) yields the following result. Proposition 2.9.
For any W and c , the fan biCamb ( W, c ) is the normal fan ofa polytope, specifically, the Minkowski sum of the generalized associahedra dual to Camb ( W, c ) and Camb ( W, c − ) . The definition of biCamb ( W, c ) seems strange a priori , but it is well-motivated a posteriori by enumerative results. The first such result is Theorem 2.10 below.When W is the symmetric group S n (i.e. when W is of type A n − ), the Coxeterdiagram of W is a path. A linear Coxeter element of S n is the product of thegenerators in order along the path. Theorem 2.10.
When W is the symmetric group S n and c is the linear Coxeterelement, the number of maximal cones in biCamb ( W, c ) is the Baxter number B ( n ) = (cid:18) n + 11 (cid:19) − (cid:18) n + 12 (cid:19) − n (cid:88) k =1 (cid:18) n + 1 k − (cid:19)(cid:18) n + 1 k (cid:19)(cid:18) n + 1 k + 1 (cid:19) . Figure 4.
The linear biCambrian fan in type A Figure 5.
The bipartite biCambrian fan in type A For more on the Baxter number, see [6, 12, 17]. Theorem 2.10 was observedempirically (in the language of lattice congruences) in [32, Section 10] and thenproven by J. West [48]. See also [22, 26]. The theorem is also related to theobservation by Dulucq and Guibert [16] that pairs of twin binary trees are countedby the Baxter number.Once one sees that the Baxter number counts maximal cones of biCamb ( W, c )for W of type A and for a particular c , it is natural to look at other types of finiteCoxeter group W , with the idea of defining a “ W -Baxter number” for each finiteCoxeter group W . Indeed, there is a good notion of a “type-B Baxter number”discovered by Dilks [15]. The Coxeter diagram of type B is also a path, and taking c to be a linear Coxeter element, the maximal cones of biCamb ( W, c ) are countedby the type-B Baxter number. Despite the nice type-B result, there seems to belittle hope for a reasonable definition of the W -Baxter number, because some typesof Coxeter diagrams are not paths and thus it is not clear how to generalize thenotion of a linear Coxeter element.There is, however, a choice of Coxeter element that can be made uniformly forall finite Coxeter groups. Since the Coxeter diagram of any finite Coxeter groupis acyclic, the diagram is in particular bipartite. Thus we can fix a bipartition S + ∪ S − of the diagram and orient each edge of the diagram from its vertex in S − to its vertex in S + . The resulting Coxeter element is called a bipartite Coxeterelement , and if c is a bipartite Coxeter element of W , we call biCamb ( W, c ) a bipartite biCambrian fan . We emphasize that the case of bipartite c is veryspecial. In particular, many of our results explicitly require that c is bipartite.Proposition 2.9 says that biCamb ( W, c ) is the normal fan of a polytope, but doesnot guarantee that this polytope is simple (equivalently, that this fan is simplicial).In fact, simpleness fails for the linear Coxeter element of S n , and this failure canbe seen already in S . (See Figure 4, and also [26, Figure 13]. The latter shows the1-skeleton of this polytope disguised as the Hasse diagram of a certain lattice.) Weconjecture that the situation is better in the bipartite case. Conjecture 2.11. If W is a bipartite Coxeter element, then biCamb ( W, c ) is asimplicial fan. (Equivalently, its dual polytope is simple.) We have verified Conjecture 2.11, with the aid of Stembridge’s packages [46], upto rank 6. Also, in Section 3.6, we prove the following theorem using alternatingarc diagrams, by appealing to some results of [13] linking the lattice theory of theweak order to the representation theory of finite-dimensional algebras, and thenapplying a folding argument.
Theorem 2.12.
Conjecture 2.11 holds in types A and B.
In Section 2.3, we will prove the following theorem.
Theorem 2.13.
If Conjecture 2.11 holds for a Coxeter group W , then the h -vector of the simplicial sphere underlying biCamb ( W, c ) , for c bipartite, has entries biNar k ( W ) . In light of the evidence for Conjecture 2.11 and in light of Theorem 2.13, wepropose the term simplicial W -biassociahedron for the polytope whose face fanis biCamb ( W, c ) for c bipartite , and simple W -biassociahedron for the polytopewhose normal fan is biCamb ( W, c ) for c bipartite . Remark 2.14.
Theorems 1.5, 2.12, and 2.13 imply that the A n -biassociahedron hasthe same h -vector as the B n -associahedron (also known as the cyclohedron). Oneis naturally led to ask whether these two polytopes are combinatorially isomorphic.The answer is no already for n = 3. The normal fan to the A -biassociahedronis shown in Figure 5. The dual graph to this fan has a vertex that is incident totwo hexagons and a quadrilateral. The graph of the B -associahedron (shown forexample in [18, Figure 3.9]) has no such vertex.2.3. The biCambrian congruence, twin sortable elements, and bisortableelements. A congruence Θ on a lattice L is an equivalence relation respectingthe meet and join operations. We now quote some combinatorial facts about latticecongruences. Proofs can be found in [38, Section 9-5]. In this paper, we consideronly finite lattices, and some results quoted in this section can fail for infinitelattices. On a finite lattice, congruences are characterized by three properties:congruence classes are intervals; the projection π Θ ↓ , mapping each element to thebottom element of its congruence class, is order preserving; and the projection π ↑ Θ ,mapping each element to the top element of its congruence class, is order preserving.The Θ-classes are exactly the fibers of π Θ ↓ . The quotient L/ Θ of a finite lattice L modulo a congruence Θ is a lattice isomorphic to the subposet induced by the set π Θ ↓ ( L ) of elements that are the bottoms of their congruence classes. The congruenceΘ is determined by the set π Θ ↓ ( L ): Specifically x ≡ y modulo Θ if and only if theunique maximal element of π Θ ↓ ( L ) below x equals the unique maximal element of π Θ ↓ ( L ) below y .The map π Θ ↓ is a lattice homomorphism from L onto the subposet π Θ ↓ ( L ), butcare must be taken to avoid misinterpreting this fact. Literally, the fact that π Θ ↓ is alattice homomorphism means that for any U ⊆ L , we have π Θ ↓ ( (cid:87) U ) = (cid:87) x ∈ U π Θ ↓ ( x )and π Θ ↓ ( (cid:86) U ) = (cid:86) x ∈ U π Θ ↓ ( x ), but in each identity, the join on the left side occurs in L while the join on the right side occurs in π Θ ↓ ( L ). It is easy to check that π Θ ↓ ( L ) isalso a join-sublattice of L , so the distinction between the join in L and the join in π Θ ↓ ( L ) is unnecessary. However, in general, π Θ ↓ ( L ) need not be a meet-sublattice of L , so in interpreting the identity π Θ ↓ ( (cid:86) U ) = (cid:86) x ∈ U π Θ ↓ ( x ), it is crucial to be clearon where the meets occur.The maximal cones of the Coxeter fan F ( W ), partially ordered according to asuitable linear functional, form a lattice isomorphic to the weak order on W . (Thisfact is true either for the right or left weak order. We will work with the right weakorder.) Any lattice congruence Θ on the weak order on W defines a fan F Θ ( W )coarsening F ( W ). (See [32, Theorem 1.1] and [32, Section 5].) Specifically, for eachΘ-class, the union of the corresponding maximal cones in F ( W ) is itself a convexcone, and the collection of all these convex cones and their faces is the fan F Θ ( W ).Each Coxeter element c specifies a congruence Θ c on the weak order called the c -Cambrian congruence . (See [33] for the definition.) The fan F Θ c ( W ) is the c -Cambrian fan Camb ( W, c ) described earlier.The set Con( L ) of all congruences on a given lattice L is itself a sublattice ofthe lattice of set partitions of L . In particular, the meet of two congruences is thecoarsest set partition of L refining both congruences. We define the c -biCambriancongruence to be the meet, in Con( W ), of the Cambrian congruences Θ c and Θ c − .The fan F Θ ( W ) for Θ = Θ c ∧ Θ c − is the coarsest common refinement of F (Θ c ( W ))and F (Θ c − ( W )). Thus the c -biCambrian fan biCamb ( W, c ) is the fan F Θ ( W ) for OXETER-BICATALAN COMBINATORICS 13
Θ = Θ c ∧ Θ c − . In particular, the c -biCambrian congruence classes are in bijectionwith the maximal cones of biCamb ( W, c ). We define the c -biCambrian lattice to be the quotient of the weak order modulo the c -biCambrian congruence. Theelements of the c -biCambrian lattice are thus in bijection with the maximal conesof biCamb ( W, c ).We write π c ↓ for the projection taking each element of W to the bottom elementof its c -Cambrian congruence class, and similarly π c − ↓ . (That is, π c ↓ stands for π Θ ↓ where Θ = Θ c .) Consider the map that sends each c -biCambrian congruence classto the pair ( π c ↓ ( w ) , π c − ↓ ( w )), where w is any representative of the class. Becausethe c -biCambrian congruence Θ is the meet Θ c ∧ Θ c − , two elements u and v are congruent in the c -biCambrian congruence if and only if π c ↓ ( u ) = π c ↓ ( v ) and π c − ↓ ( u ) = π c − ↓ ( v ). Thus, the map from classes to pairs is a well-defined bijectionfrom c -biCambrian congruence classes to its image.The bottom elements of the c -Cambrian congruence are called c -sortable ele-ments . (In fact c -sortable elements have an independent combinatorial definition[34, Section 2], but were shown to be the bottom elements of c -Cambrian congru-ences in [35, Theorems 1.1 and 1.4].) Given elements u and v of W , we define thepair ( u, v ) to be a pair of twin ( c, c − ) -sortable elements of W if there exists w ∈ W such that u = π c ↓ ( w ) and v = π c − ↓ ( w ). The map considered in the pre-vious paragraph is a bijection between c -biCambrian congruence classes and pairsof twin ( c, c − )-sortable elements of W . The twin sortable elements are similar inspirit to the twin binary trees of [16], which were already mentioned in connectionwith Theorem 2.10. Indeed, for W of type A and c linear, the connection is implicitin the construction in [26] of a diagonal rectangulation from a pair of binary trees.(See also [26, Remark 6.6].) Also in type A, but for general c , the twin binary treesare generalized in [11] to twin Cambrian trees , which correspond explicitly topairs of twin ( c, c − )-sortable elements. Indeed, [11, Proposition 36] amounts toanother computation of the type-A biCatalan number, quite different from the twogiven here (in Sections 2.1 and 3.4).Another set of objects naturally in bijection with c -biCambrian congruenceclasses are the bottom elements of c -biCambrian congruence classes. We coin theterm c -bisortable elements for these bottom elements. Although the c -sortableelements have a direct combinatorial characterization [34, Section 2], we currentlyhave no direct combinatorial characterization of c -bisortable elements. We do offerthe following indirect characterization of c -bisortable elements in terms of c -sortableelements and c − -sortable elements. Proposition 2.15.
For any c , an element w ∈ W is c -bisortable if and only if thereexists a c -sortable element u and a c − -sortable element v such that w = u ∨ v inthe weak order. When w is c -bisortable, we can take u = π c ↓ ( w ) and v = π c − ↓ ( w ) .Proof. Given c -bisortable w , take u = π c ↓ ( w ) and v = π c − ↓ ( w ). Then u ≤ w and v ≤ w . Since Cambrian congruence classes are intervals, any upper bound w (cid:48) for u and v with w (cid:48) ≤ w is congruent to u modulo Θ c and congruent to v modulo Θ c − .Thus w (cid:48) is congruent to w in the c -biCambrian congruence. Since w is the bottomelement of its c -biCambrian congruence class, we conclude that w (cid:48) = w . We haveshown that w = u ∨ v .Suppose w = u ∨ v for some c -sortable element u and some c − -sortable element v .Since π c ↓ ( w ) is the unique maximal c -sortable element below w , we have π c ↓ ( w ) ≥ u . Similarly, π c − ↓ ( w ) ≥ v . If there exists w (cid:48) < w in the same c -biCambrian congruenceclass as w , then w (cid:48) ≥ π c ↓ ( w (cid:48) ) = π c ↓ ( w ) ≥ u and w (cid:48) ≥ π c − ↓ ( w (cid:48) ) = π c − ↓ ( w ) ≥ v . Thiscontradicts the fact that w = u ∨ v , and we conclude that w is c -bisortable. (cid:3) Recall that for any congruence Θ on a finite lattice L , the set π Θ ↓ ( L ) is a join-sublattice of L . The Cambrian congruences have a stronger property: For anyCoxeter element c , the c -sortable elements constitute a sublattice [35, Theorem 1.2]of the weak order on W . It is natural to ask whether the same is true for c -bisortableelements, but the answer is no. We give an example for W = S and bipartite c :The permutations 45312 and 53142 are both c -bisortable but their meet 31452 isnot. (To check this example, Proposition 3.6 will be very helpful.)Each c -bisortable element v covers some number of elements in the c -biCambrianlattice. By a general fact on lattice quotients (see for example [36, Proposi-tion 6.4]), v covers the same number of elements in the weak order on W . Thisnumber is des( v ), the number of descents of v . (We will define descents in Sec-tion 4.5.) The descent generating function of c -bisortable elements is thesum (cid:80) x des( v ) over all c -bisortable elements v . For bipartite c , its coefficients arethe W -biNarayana numbers. A general fact about lattice quotients of the weakorder [32, Proposition 3.5] implies that, when biCamb ( W, c ) is simplicial, thedescent generating function of c -bisortable elements equals the h -polynomial of biCamb ( W, c ). In the bipartite case, Theorem 2.13 follows immediately.2.4.
Twin clusters and bicluster fans.
Clusters of almost positive roots wereintroduced in [20], where they were used to define generalized associahedra. In [21],clusters of almost positive roots were used to model cluster algebras of finite type.Here, we will not need the cluster-algebraic background, which can be found in[21]. Instead, we define almost positive roots and c -compatibility and quote someresults about c -clusters and their relationship to c -sortable elements. We will alsonot need the more refined notion of “compatibility degree.”In a finite root system, the almost positive roots are those roots which eitherare positive, or are the negatives of simple roots. The definition of compatibility in[20] is a special case (namely the bipartite case) of what we here call c -compatibility.The general definition was given in [27], but here we give a rephrasing found in [34,Section 7], translated into the language of almost positive roots.We write { α , . . . , α n } for the simple roots and { s , . . . , s n } for the simple re-flections. For each i in { , . . . , n } , we define an involution σ i on the set of almostpositive roots by(2.1) σ i ( β ) := (cid:26) β if β = − α j with j (cid:54) = i, or s i β otherwise . We write [ β : α i ] for the coefficient of α i in the expansion of β in the basis of simpleroots. A simple reflection s i is initial in a Coxeter element c if c has a reducedword starting with s i . If s i is initial in c , then s i cs i is another Coxeter element.The c -compatibility relations are a family of symmetric binary relations (cid:107) c onthe almost positive roots. They are the unique family of relations with(i) For any i in { , . . . , n } , and Coxeter element c , − α i (cid:107) c β if and only if [ β : α i ] = 0 . OXETER-BICATALAN COMBINATORICS 15 (ii) For each pair of almost positive roots β and β , each Coxeter element c ,and each s i initial in c , β (cid:107) c β if and only if σ i ( β ) (cid:107) s i cs i σ i ( β ) . The c -clusters are the maximal sets of pairwise c -compatible almost positiveroots. By [20, Theorem 1.8] and [27, Proposition 3.5], for fixed W , all c -clusters areof the same size, and furthermore, each is a basis for the root space (the span ofthe roots). Write R ≥ C for the nonnegative linear span of a c -cluster C . Then [20,Theorem 1.10] and [27, Theorem 3.7] state that the cones R ≥ C , for all c -clusters C ,are the maximal cones of a complete simplicial fan. We call this fan the c -clusterfan .We define the c -bicluster fan to be the coarsest common refinement of the c -cluster fan and its antipodal opposite. A pair ( C , C ) of c -clusters is called a pairof twin c -clusters if the cones R ≥ C and − R ≥ C (the nonpositive linear spanof C ) intersect in a full-dimensional cone. It is immediate that maximal cones inthe c -bicluster fan are in bijection with pairs of twin c -clusters. Example 2.16.
For W of type A , up to symmetry there are two different c -bicluster fans: one for linear c and one for bipartite c , shown in Figures 6 and 7respectively. These are again stereographic projections as explained in Example 2.6.The two c -bicluster fans in Example 2.16 are combinatorially isomorphic. Despitethis tantalizing fact, in this paper, we only explore bicluster fans in the special caseof bipartite Coxeter elements (the original setting of [20, 21]), where they are easilyrelated to biCambrian fans. For the bipartite choice of c , [39, Theorem 9.1] saysthat the c -Cambrian fan is linearly isomorphic to the cluster fan. Combining thisfact with Proposition 2.8, we have the following theorem. Theorem 2.17.
For all finite Coxeter groups W and for bipartite c , the c -biclusterfan is linearly isomorphic to the c -biCambrian fan. Remark 2.18.
We emphasize that Theorem 2.17 requires the hypothesis that c isbipartite. In contrast, when W is of type A and c is the linear Coxeter element,the c -bicluster fan and the c -biCambrian fan don’t even have the same number ofregions.Because of the bijection between c -bisortable elements and maximal cones in biCamb ( W, c ) and the bijection between maximal cones in the c -bicluster fan andpairs of twin c -clusters, we have the following immediate consequence of Theo-rem 2.17. Theorem 2.19.
For all finite Coxeter groups W , c -bisortable elements for bipar-tite c are in bijection with pairs of twin c -clusters. Combining Theorems 2.13 and 2.17, we obtain the following theorem.
Theorem 2.20.
If Conjecture 2.11 holds for a Coxeter group W , then the bipartite c -bicluster fan is simplicial and the h -vector of the underlying simplicial sphere hasentries biNar k ( W ) . Twin noncrossing partitions.
The absolute order on a finite Coxeter group W is the prefix order (or equivalently the subword order) on W relative to thegenerating set T , the set of reflections in W . (By contrast, the prefix order relativeto the simple reflections S is the weak order, while the subword order relative to α − α α − α α − α Figure 6.
The linear bicluster fan in type A α − α α − α α − α Figure 7.
The bipartite bicluster fan in type A S is the Bruhat order.) We will use the symbol ≤ T for the absolute order. The c -noncrossing partitions in a finite Coxeter group W are the elements of W contained in the interval [1 , c ] T in the absolute order on W . For details on theabsolute order and noncrossing partitions, see for example [5, Chapter 2]. For ourpurposes, the key fact is a theorem of Brady and Watt.Let W be a finite Coxeter group of rank n represented as a reflection group in R n and let T be the set of reflections of W . For each reflection t ∈ T , let β t be thecorresponding positive root. Given w ∈ [1 , c ] T , define a cone F c ( w ) = (cid:110) x ∈ R n : x · β t ≤ ∀ t ≤ T w, x · β t ≥ ∀ t ≤ T cw − (cid:111) . The following theorem combines [9, Theorem 1.1] with [9, Theorem 5.5].
Theorem 2.21.
For c bipartite, the map F c is a bijection from [1 , c ] T to the set ofmaximal cones in the c -Cambrian fan. The astute reader will notice a difference between our definition of F c and thedefinition appearing in [9, Section 1]. The set of reflections t such that t ≤ T w is theintersection of T with some (not necessarily standard) parabolic subgroup of W .The definition in [9] imposes inequalities x · β t ≤ β t that are simpleroots for that parabolic subgroup. Our definition imposes additional inequalities,all of which are implied by the inequalities for the simple roots. We similarly addadditional redundant inequalities of the form x · β t ≥ u, v ∈ [1 , c ] T ,we call ( u, v ) a pair of twin c -noncrossing partitions if F c ( u ) ∩ ( − F c ( v )) isfull-dimensional. Similarly, given u ∈ [1 , c ] T and v ∈ [1 , c − ] T , we call ( u, v ) a pairof twin ( c, c − ) -noncrossing partitions if F c ( u ) ∩ F c − ( v ) is full-dimensional.Theorem 2.21 now immediately implies the following theorem. Theorem 2.22.
For all W and bipartite c , the c -bisortable elements are in bi-jection with pairs of twin c -noncrossing partitions and with pairs of twin ( c, c − ) -noncrossing partitions. Bipartite c -bisortable elements and alternating arc diagrams In this section, we show how bipartite c -bisortable elements of type A are inbijection with certain objects called alternating arc diagrams. We then prove thetype-A enumeration of bipartite c -bisortable elements in Theorem 1.1 by countingalternating arc diagrams and prove the type-B enumeration by counting centrallysymmetric alternating arc diagrams.3.1. Pattern avoidance.
The Coxeter group of type A n is the symmetric group S n +1 . We will write permutations x in S n +1 in their one-line notations x · · · x n +1 .In the weak order on permutations in S n +1 , there is a cover x · · · x n +1 < · y · · · y n +1 if and only if there exists i such that y i = x i +1 > x i = y i +1 and y j = x j for j (cid:54)∈ { i, i + 1 } . We say that x is covered by y via a swap in positions i and i + 1.The Cambrian congruences on S n +1 are described in detail in [33]. We quotepart of the description here. The simple generator s i for A n is the transposition( i i +1), for i = 1 , , . . . n . Each Coxeter element c can be encoded by a coloring ofthe elements 2 , . . . , n that we call a barring . Each element i is either overbarred and marked i if s i occurs before s i − in every reduced word for c , or underbarred and marked i if s i occurs after s i − in every reduced word for c . Passing from c to c − means swapping overbarring with underbarring.We say x is obtained from y by a 231 → move if x is covered by y via aswap in positions i and i + 1, for some i , and if there exists an overbarred element x j with j < i and x i < x j < x i +1 . Similarly, x is obtained from y by a 312 → move if x is covered by y via a swap in positions i and i + 1, for some i , and if thereexists an underbarred element x j with i + 1 < j and x i < x j < x i +1 . Combining[33, Proposition 5.3] and [33, Theorem 6.2], we obtain the following proposition: Proposition 3.1.
Suppose x and y are permutations in S n +1 with x < · y in theweak order, and assume that the numbers , . . . , n have been barred according to c .Then x and y are in the same c -Cambrian congruence class if and only if x isobtained from y by a → move or a → move. As an immediate corollary, we see that a permutation y is the bottom elementof its c -Cambrian congruence class (i.e. is c -sortable) if and only if none of thepermutations covered by y are obtained from y by a 231 →
213 move or a 312 → bca of y with a < b < c , with c immediately preceding a , and with b overbarred and no subsequence cab of y with a < b < c , with c immediately preceding a , and with b underbarred. In this case,we say that y avoids
231 and 312.We can similarly describe bottom elements of c -biCambrian congruence classes(the c -bisortable elements), keeping in mind that passing from c to c − meansswapping overbarring with underbarring: An element y is the bottom element ofits c -biCambrian congruence class if and only if none of the permutations covered by y are obtained from y by a 231 →
213 or 312 →
132 move that is also a 231 → →
132 move. (Compare [11, Remark 34].) For c linear, the c -bisortablepermutations are the twisted Baxter permutations of [26, Section 4.2]. In general, c -bisortable permutations may be described by a complicated pattern-avoidancecondition that we will only describe, in Propositions 3.6 and 3.7, for the case ofbipartite c , where it becomes much simpler.3.2. Noncrossing arc diagrams.
We now review the notion of noncrossingarc diagrams from [37]. Beginning with n + 1 distinct points on a vertical line,numbered 1 , . . . , n + 1 from bottom to top, we draw some (or no) curves called arcs connecting the points. Each arc moves monotone upwards from one of thepoints to another, passing either to the left or to the right of each point in between.Furthermore no two arcs may intersect in their interiors, no two arcs share the sameupper endpoint, and no two arcs may share the same lower endpoint. We considerarc diagrams only up to their combinatorics, i.e. which pairs of points are joinedby an arc and which points are left and right of each arc.Given a permutation x · · · x n +1 in S n +1 , we define a noncrossing arc diagram δ ( x · · · x n +1 ). Each descent x i > x i +1 becomes an arc α in δ ( x · · · x n +1 ) withlower endpoint x i +1 and upper endpoint x i . For each integer j with x i +1 < j < x i that occurs to the left of x i in x · · · x n +1 , the point j is left of the arc α . For eachinteger j with x i +1 < j < x i that occurs to the right of x i +1 in x · · · x n +1 , thepoint j is right of the arc α . It was shown in [37, Theorem 3.1] that δ is a bijectionfrom permutations to noncrossing arc diagrams. More specifically, for each k , themap δ restricts to a bijection from permutations with k descents to noncrossing arcdiagrams with k arcs. OXETER-BICATALAN COMBINATORICS 19 A c -sortable arc is an arc that belongs to δ ( v ) for some c -sortable permuta-tion v . The following characterization of c -sortable arcs in terms of the barringassociated to c is immediate from the pattern-avoidance description above. (Com-pare [37, Example 4.9].) Proposition 3.2.
For W = A n and any c , the c -sortable arcs are the arcs that donot pass to the left of any underbarred element of { , . . . , n } and do not pass to theright of any overbarred element of { , . . . , n } . In particular, since c and c − correspond to opposite barrings, the only arcs thatare both c and c − -sortable are the arcs that connect adjacent endpoints i and i +1.(This is a restatement of the type-A case of Remark 2.7 in terms of noncrossing arcdiagrams.)Combining the above descriptions of c -sortable and c -bisortable elements in termsof overbarred and underbarred elements, we obtain the following proposition. Proposition 3.3.
For W = A n and any c , the map δ restricts to a bijection from c -bisortable permutations with k descents to noncrossing arc diagrams on n + 1 vertices with k arcs, each of which is either c or c − -sortable.Proof. Suppose x = x · · · x n is a permutation such that δ ( x ) has an arc that isneither c -sortable nor c − -sortable. This arc has upper endpoint x i and lowerendpoint x i +1 for some i and it fails the conclusion of Proposition 3.2 for c and for c − . That is, it either passes left of an underbarred element or right of an overbarredelement and it either passes left of an overbarred element or right of an underbarredelement. Thus, switching x i with x i +1 is both a 231 →
213 or 312 →
132 move and a 231 →
213 or 312 →
132 move. Therefore, x is not c -bisortable. The argument iseasily reversed to prove the converse. (cid:3) Alternately, Proposition 3.3 follows from the description of the c -biCambriancongruence as the meet of the c -Cambrian and c − -Cambrian congruences.3.3. Alternating arc diagrams.
We now consider the case where c is bipartite.Let c + be the product of the simple generators s i where i is even, and c − be theproduct of the simple generators s i where i is odd. The bipartite Coxeter elements in A n are c + c − and its inverse c − c + . The barring of the numbers 2 , . . . , n associated to c + c − has all even numbers overbarred and all odd numbers underbarred. A right-even alternating arc is an arc that passes to the right of even vertices and to theleft of odd vertices. A left-even alternating arc is an arc that passes to the leftof even vertices and to the right of odd vertices. A right-even alternating arcdiagram is a noncrossing arc diagram all of whose arcs are right-even alternating,and left-even alternating arc diagrams are defined analogously. The followingproposition is an immediate consequence of Proposition 3.2. Proposition 3.4.
Suppose W = A n and c is the bipartite Coxeter element c + c − .(1) The map δ restricts to a bijection from c -sortable permutations to right-evenalternating arc diagrams.(2) The map δ restricts to a bijection from c − -sortable permutations to left-even alternating arc diagrams.In each case, δ restricts further to send permutations with k descents bijectively toarc diagrams with k arcs. Figure 8.
Some alternating noncrossing arc diagramsAn alternating arc is an arc that is either right-even alternating or left-evenalternating or both. We call a noncrossing arc diagram consisting of alternatingarcs an alternating arc diagram . Figure 8 shows several alternating noncrossingarc diagrams. From left to right, they are δ (5371624), δ (4631275), and δ (4275136).The following proposition is the bipartite case of Proposition 3.3. Proposition 3.5.
For W = A n and c bipartite, the map δ restricts to a bijectionfrom c -bisortable permutations with k descents to alternating arc diagrams on n + 1 points with k arcs. Observe that an arc fails to be alternating if and only if it passes on the same sideof two consecutive numbers. Thus, we obtain the following simpler description ofthe pattern avoidance condition defining bipartite c -bisortable elements. (Compare[11, Remark 34].) Proposition 3.6. If c is the bipartite Coxeter element c + c − of A n , a permutation x = x · · · x n +1 is c -bisortable if and only if, for every descent x i > x i +1 , thereexists no k with x i +1 < k < k + 1 < x i such that k and k + 1 are on the same sideof the descent (i.e. k and k + 1 both left of x i or both right of x i +1 ). The condition in Proposition 3.6 is that x avoids subsequences dabc , dacb , bcda ,and cbda with a < b < c < d , with d and a adjacent in position , and with b and c being adjacent in value . This is an instance of bivincular pattern avoidance inthe sense of [8, Section 2]. We will not review the notation for bivincular patternsfrom [8], but we restate Proposition 3.6 in that notation as follows: Proposition 3.7.
For c bipartite, a permutation is c -bisortable if and only if itavoids the bivincular patterns (2341 , { } , { } ) , (3241 , { } , { } ) , (4123 , { } , { } ) ,and (4132 , { } , { } ) . Counting alternating arc diagrams.
Let [ n ] denote the set { , , . . . , n } .To prove the type-A enumeration of bipartite c -bisortable elements in Theorem 1.1,we give a bijection π from noncrossing alternating arc diagrams on n + 1 verticeswith k arcs to pairs ( S, T ) of subsets of [ n ] with | S | = | T | = k .Suppose that Σ is an alternating arc diagram. Whenever we encounter a right-even alternating arc in Σ with endpoints i < j , we put i into S and j − T ;whenever we encounter a left-even alternating arc with endpoints i < j we put j − S and i into T . More precisely, suppose that Σ is an alternating arc diagramwith k arcs. Let S (cid:48) denote the set of numbers i such that i is bottom endpoint ofa right-even alternating arc in Σ and let S (cid:48)(cid:48) denote the set of numbers j − j is the top endpoint of a left-even alternating arc in Σ. Let T (cid:48) denote the set OXETER-BICATALAN COMBINATORICS 21 of numbers j (cid:48) − j (cid:48) is the top endpoint of a right-even alternating arc inΣ and let T (cid:48)(cid:48) denote the set of numbers i (cid:48) such that i (cid:48) is the bottom endpoint of aleft-even alternating arc. The map π sends Σ to the pair ( S (cid:48) ∪ S (cid:48)(cid:48) , T (cid:48) ∪ T (cid:48)(cid:48) ). Theorem 3.8.
The map π is a bijection from the set of alternating arc diagramson n + 1 points to the set of pairs of subsets of [ n ] of the same size. For each k , thebijection restricts to a bijection from alternating arc diagrams with k arcs to pairsof subsets of size k . In preparation for the proof of Theorem 3.8, we will break each alternatingdiagram into smaller pieces. Two alternating arcs with endpoints i < j and i (cid:48) < j (cid:48) overlap if the intersection of the sets { i, . . . , j − } and { i (cid:48) , . . . , j (cid:48) − } is nonempty.Informally, the arcs overlap if some part of one arc passes alongside of the other arc.(If they only touch at their endpoints but don’t pass alongside one another, thenthey do not overlap). Given a collection E of arcs, we can define an “overlap graph”with vertices E and edges given by overlapping pairs in E . We say that the collection E is overlapping if this overlap graph is connected. Each noncrossing arc diagramcan be broken into overlapping components, maximal overlapping collections ofarcs. The definition of alternating arc diagrams and the definition of right-evenand left-even alternating arcs let us immediately conclude that two distinct arcsappearing in the same alternating arc diagram, one right-even alternating and oneleft-even alternating, cannot overlap. We have proved the following fact. Proposition 3.9.
Each overlapping component of an alternating arc diagram fitsexactly one of the following descriptions: (1) It consists of right-even alternatingarcs that are not left-even alternating; (2) It consists of left-even alternating arcsthat are not right-even alternating; or (3) it consists of a single arc that is right-evenand left-even alternating (and thus connects two adjacent points).
Proposition 3.9 implies that, on each overlapping component, the map π collectsall of the top endpoints of the arcs into one set, and all of the bottom endpointsinto the other set.Now we describe how to break an alternating diagram Σ into its overlappingcomponents. Let P (Σ) be the set of numbers p ∈ [ n + 1] such that no arc inΣ passes left or right of p . (A point p ∈ P (Σ) may still be an endpoint of oneor two arcs.) Write P (Σ) = { p , . . . , p m } with p < · · · < p m . In every case, p = 1 and p m = n + 1. For each i , we claim that an arc in Σ has its lowerendpoint in { p i − , p i − + 1 , . . . , p i − } if and only if it has its upper endpoint in { p i − + 1 , p i − + 2 , . . . , p i } . Indeed, if an arc has a lower endpoint in { p i − , p i − +1 , . . . , p i − } , then since it cannot pass on either side of p i , it must end at anumber in the set { p i − + 1 , p i − + 2 , . . . , p i } . A similar argument proves theconverse, so we have established the claim. Let Σ i denote the set of arcs withlower endpoints in { p i − , p i − + 1 , . . . , p i − } (and thus with upper endpoints in { p i − + 1 , p i − + 2 , . . . , p i } ). By construction, Σ i is an overlapping component, andall overlapping components are Σ i for some i . Let ( S i , T i ) be the image of Σ i under π , so that π (Σ) = ( (cid:83) mi =1 S i , (cid:83) mi =1 T i ).We say that two arcs are compatible if there is a noncrossing arc diagramcontaining both arcs. Our next task is to understand for which pairs ( s, t ) and( s (cid:48) , t (cid:48) ) there exists an overlapping pair of compatible alternating arcs , one withendpoints s and t + 1 and one with endpoints s (cid:48) and t (cid:48) + 1. Since the arcs mustoverlap but may not share the same bottom endpoint and may not share the same top endpoint, and taking without loss of generality s < s (cid:48) , there are only two cases.These cases are covered by the following two lemmas, which are easily verified. Lemma 3.10.
Suppose s < s (cid:48) ≤ t < t (cid:48) . Then there exist two compatible alternatingarcs, one with endpoints s and t + 1 and one with endpoints s (cid:48) and t (cid:48) + 1 if andonly if s (cid:48) and t have the same parity. The pair of arcs can be chosen in exactlytwo ways, either both as right-even alternating arcs or both as left-even alternatingarcs. Lemma 3.11.
Suppose s < s (cid:48) < t (cid:48) < t . Then there exist two compatible alternatingarcs, one with endpoints s and t + 1 and one with endpoints s (cid:48) and t (cid:48) + 1 if andonly if s (cid:48) and t (cid:48) have opposite parity. The pair of arcs can be chosen in exactlytwo ways, either both as right-even alternating arcs or both as left-even alternatingarcs. Given a pair (
S, T ) of k -subsets of [ n ], we will always write S = { s , . . . , s k } with s < · · · < s k and T = { t , . . . , t k } with t < · · · < t k . Define Q ( S, T ) to be theset of numbers q ∈ [ n + 1] such that, for all j from 1 to k , neither s j < q ≤ t j , nor t j < q ≤ s j . Lemma 3.12.
Let Σ be an alternating arc diagram. Then Q ( π (Σ)) = P (Σ) .Proof. Write (
S, T ) for π (Σ). If p ∈ P (Σ), then no arc passes left or right of p .Thus there exists k such that s j and t j are less than p for all j ≤ k and s j and t j are greater than or equal to p for all j > k . We see that p ∈ Q ( S, T ).Suppose that q ∈ Q ( S, T ), and there exists some arc α that passes to the leftor right of q . The arc α belongs to some overlapping component of Σ, and eachpair s i , t i in the image of a different component satisfies s i , t i < q or s i , t i > q .Thus, we may as well assume that Σ consists of a single overlapping component.Write π (Σ) = ( { s , . . . , s k } , { t , . . . , t k } ) with s < · · · < s k and t < · · · < t k .Lemma 3.9 says that Σ consists of either right-even overlapping arcs or left-evenoverlapping arcs. Without loss of generality, we assume that Σ consists of onlyright-even overlapping arcs, so that { s , . . . , s k } is the set of bottom endpoints ofthose arcs. Thus, s i ≤ t i for each i = 1 , , . . . , k . Let s i be the bottom endpointof α , and let l be the largest number such that s l < q . We make two observations.First, α must connect s i with t j + 1, where j is strictly greater than i (otherwise s i < q ≤ t j ≤ t i ), and j is strictly greater than l (otherwise s j < q ≤ t j ). Second, t l +1 ≥ q > t l , because t l +1 ≥ s l +1 ≥ q > t l ≥ s l . We conclude that each numberin the set of bottom endpoints { s l +1 , s l +2 , . . . , s k } must connect with a number inthe set { t l +1 + 1 , . . . , t k + 1 } . Since t j + 1 is already connected to s i , there is somenumber in the set { t l +1 + 1 , . . . , t k + 1 } that is the top endpoint of two arcs, andthat is a contradiction. (cid:3) We are now prepared to prove the main theorem of this section.
Proof of Theorem 3.8.
We first show that π is well-defined. Since each arc in Σcontributes exactly one of its endpoints to S (cid:48) ∪ S (cid:48)(cid:48) and the other to T (cid:48) ∪ T (cid:48)(cid:48) , both S (cid:48) ∪ S (cid:48)(cid:48) and T (cid:48) ∪ T (cid:48)(cid:48) have size k as long as each contribution to S (cid:48) ∪ S (cid:48)(cid:48) is distinctand each contribution to T (cid:48) ∪ T (cid:48)(cid:48) is distinct. Each contribution to S (cid:48) is distinctbecause no two arcs share the same lower endpoint, and each contribution to S (cid:48)(cid:48) is distinct because no two arcs share the same upper endpoint. Proposition 3.9implies that a right-even alternating arc with bottom endpoint i and a distinct OXETER-BICATALAN COMBINATORICS 23 left-even alternating arc with top endpoint i + 1 are not compatible. Thus theonly elements of S (cid:48) ∩ S (cid:48)(cid:48) come from arcs that are both right-even alternating andleft-even alternating, and we see that each contribution to S (cid:48) ∪ S (cid:48)(cid:48) is distinct. Thesymmetric argument shows that each contribution to T (cid:48) ∪ T (cid:48)(cid:48) is distinct. We haveshown that π is a well-defined map from alternating arc diagrams with k arcs topairs of k -element subsets of [ n ].We complete the proof by exhibiting an inverse η to π . Let ( S, T ) be a pairof k -element subsets of [ n ]. Write Q ( S, T ) = { q , . . . , q m } with q < · · · < q m .For each i from 1 to m , define S i = S ∩ { q i − , q i − + 1 , . . . , q i − } and T i = T ∩ { q i − , q i − + 1 , . . . , q i − } . We claim that | S i | = | T i | , and more specifically,that s j ∈ S i if and only if t j ∈ T i . Indeed, suppose s j ∈ S i , so that q i − ≤ s j < q i .If t j < q i − , then t j < q i − ≤ s j , contradicting the fact that q i − ∈ Q ( S, T ). If t j ≥ q i , then s j < q i ≤ t j , contradicting the fact that q i ∈ Q ( S, T ). We concludethat t j ∈ T i . The symmetric argument completes the proof of the claim.Now, in light of Lemma 3.12 and the definition of π , by subtracting q i − − S i and T i , we reduce to the case where m = 1 and thus Q = { , n + 1 } and ( S , T ) = ( S, T ). In particular, all of the arcs in the diagram η ( S, T )are right-even alternating, or all of the arcs are left-even alternating. If n = 1, theneither ( S, T ) = ( ∅ , ∅ ), in which case η ( S, T ) has no arc, or (
S, T ) = ( { } , { } ), inwhich case η ( S, T ) has an arc connecting 1 and 2.If n >
1, then we observe that the element 1 must be in S or in T but mustnot be in both. Indeed, if 1 is in neither set or in both, we see that 2 ∈ Q ( S, T ),and this is a contradiction. In particular, we will need to construct an arc whoselower endpoint is 1 and whose upper endpoint is above 2. This arc will pass by2, and so it is either right-even alternating or left-even alternating (but not both).If 1 ∈ S , then the corresponding arc is right-even alternating, and if 1 ∈ T thisarc is left-even alternating. Without loss of generality, we assume 1 ∈ S , so thateach i in S is a bottom endpoint and for each j in T , j + 1 is a top endpoint of aright-even alternating arc in η ( S, T ). To complete the proof, we show that there isa unique way to pair off each bottom endpoint in S with a top endpoint in T sothat the union of the resulting arcs is a noncrossing arc diagram. Since the arcs inthe diagram are all right-even alternating, we must pair each element of S with alarger element of T .We first decide which element of T we should pair with s k . Because s k is themaximum element of S , Lemma 3.10 implies that we must pair s k with some t (cid:48) such that (cid:8) t ∈ T : s k < t < t (cid:48) , t − s k odd (cid:9) is empty. Similarly, Lemma 3.11 impliesthat we must either pair s k with t k or pair s k with some t (cid:48) such that t (cid:48) − s k isodd. Furthermore, if we choose t (cid:48) according to those two rules, no matter howwe pair the remaining elements of S and T , the arcs produced will be compat-ible with the arc whose bottom endpoint is s k . We are forced to pair s k withmin { t ∈ T : t ≥ s k , t − s k odd } , or with t k if { t ∈ T : t ≥ s k , t − s k odd } = ∅ . Byinduction on k , there is a unique way to pair the elements of S \ { s k } with theelements of T \ (cid:8) t (cid:48) (cid:9) to make a noncrossing alternating diagram. Putting in the pair( s k , t (cid:48) ) we obtain the unique pairing of elements of S with elements of T to make anoncrossing alternating diagram. The base of the induction is where k = 1. Hereexistence of a pairing is trivial and uniqueness comes from the requirement thatthe arc whose bottom endpoint is 1 must be right-even alternating. (cid:3) Figure 9.
The indecomposable right-even alternating arc dia-grams on 3, 4, or 5 pointsThe proof of Theorem 3.8 completes the proof of Theorem 1.3 and Theorem 1.5for type A.
Remark 3.13.
The proof of Theorem 3.8 provides a key insight that leads to ourproof of Theorems 1.2 and 1.3: An alternating arc diagram decomposes into disjointpieces such that each piece is either right-even alternating or left-even alternating.We now describe a way of counting alternating arc diagrams by decomposing intoleft-even and right-even pieces (more coarsely than the decomposition into overlap-ping collections of arcs used in the proof of Theorem 3.8).Recall that an arc is both right-even alternating and left-even alternating if andonly if it connects consecutive points. We call such an arc a simple arc (and everyother arc is called non-simple ). Given an alternating arc diagram Σ, let R be theset of points p such that there exists a non-simple right-even alternating arc α in Σsuch that α passes alongside p , or α has p as an endpoint. Similarly, let L be theset of p (cid:48) satisfying the above but with α left-even alternating.We will refer to a set of integers of the form { a, a + 1 , . . . , b − , b } as an interval .Let R , . . . , R k be the maximal intervals contained in R , so that in particular R isa disjoint union of the R i , and any two of the R i have at least one point betweenthem that is not in R . Thus on each R i , we have an “indecomposable piece” ofthe diagram for Σ. See Figure 9 for the right-even indecomposable pieces on 3, 4,or 5 points. (The restriction of Σ to each R i is a union of overlapping collectionsof arcs, in the sense of Proposition 3.9. Each overlapping collection of arcs liveson an interval contained in R i , and these intervals are pairwise disjoint exceptfor intersecting at their endpoints.) Symmetrically, L breaks into an analogouscollection of indecomposable pieces consisting of left-even alternating arcs. Sincethe arcs of Σ don’t cross, each R i is disjoint from each L j , except possibly at theirendpoints. The enumeration of alternating arc diagrams can be decomposed as asum over all choices of R and L and their decompositions into pieces R , . . . , R k and L i , . . . , L m . Each term in the sum is a product of: a power of 2; a factor foreach R i equal to the number of indecomposable pieces that can be constructed thatinterval; and an analogous factor for each L j . The power of 2 arises because thereare “gaps” between intervals of R or L where we can fill in simple arcs or not. Ourproof of Theorems 1.2 and 1.3 generalizes this method, which can be carried outuniformly for all finite Coxeter groups.Each indecomposable diagram with right-even alternating arcs corresponds (via δ )to a c -sortable permutation whose set of cover reflections has no simple reflectionsand also has full support. (Cover reflections and supports will be defined in Sec-tion 4.3 for general Coxeter groups. The cover reflections of a permutation are thetranspositions ( i j ) such that i immediately precedes j and i > j . The supportof an element is the set of simple reflections appearing in a reduced word for the OXETER-BICATALAN COMBINATORICS 25 element. The requirement on supports here is that the union of the supports ofthe cover reflections is full.) Similarly, each indecomposable diagram with left-evenalternating arcs corresponds to a c − -sortable permutation whose set of cover re-flections has no simple reflections and also has full support. Thus the proof breaksbipartite biCambrian objects (the alternating arc diagrams) into pieces that belongto ordinary Catalan combinatorics. More specifically, once we fix the type of arc(right-even or left-even alternating), the number of indecomposable diagrams on m + 1 points is the number that in Section 4 will be called the double-positiveCatalan number Cat ++ ( A m ).In the general setting, the role of the arcs in an alternating arc diagram is playedby the canonical joinands of a bipartite c -bisortable element. (The latter are join-irreducible c - or c − -sortable elements, and are defined in Section 4.3. For theconnection between arcs and canonical join representations, see [37, Section 3].)The fact that distinct right-even alternating and left-even alternating arcs maynot overlap is a special case of the following fact, which we will prove uniformlyin Section 4.5: Suppose c is bipartite and w is a c -bisortable element with a c -sortable canonical joinand u and a c − -sortable canonical joinand v . If u = v ,then they equal a simple reflection, and if u (cid:54) = v , then the supports of u and v aredisjoint. Therefore, we can partition the set of canonical joinands of a c -bisortableelement into a set of simple reflections, a set of non-simple c -sortable join-irreducibleelements, and a set of non-simple c − -sortable join-irreducible elements. The set ofnon-simple c -sortable canonical joinands is a collection of “indecomposable pieces”that belong to ordinary Catalan combinatorics. Just as the set R broke into adisjoint union of the intervals R , . . . , R k , we break up this collection on c -sortablecanonical joinands as follows: An “indecomposable piece” corresponds to a subsetof these canonical joinands that has full support on some irreducible parabolicsubgroup of W . The number of possible pieces for each standard parabolic subgroupis a double-positive Catalan number, so we obtain a formula counting c -bisortableelements in terms of double-positive Catalan numbers. We complete the proof byshowing that the same formula also counts antichains in the doubled root poset. Remark 3.14.
Looking ahead to Section 4, the previous remark implies an in-terpretation of the type-A double-positive Narayana number which—after somecombinatorial manipulations that amount to changing from a bipartite Coxeterelement to a linear Coxeter element—coincides with the interpretation given in[3, Theorem 1.1].3.5.
Enumerating bipartite c -bisortable elements in type B. In this section,we use certain alternating arc diagrams to prove the enumeration of bipartite c -bisortable elements of type B given in Theorem 1.1. In order to reuse much of ourwork from Section 3.4, we realize the weak order on B n as a sublattice of the weakorder on A n − , through the usual signed permutation model.Let x = x − n . . . x − x . . . x n be a permutation of {± , ± , . . . , ± n } . Recall that x is a signed permutation if x i = − x − i for each i ∈ [ n ]. A signed permutationis completely determined by its abbreviated notation x x · · · x n . We refer to thelonger sequence x − n . . . x − x . . . x n as the full one-line notation for x . The weakorder on B n is isomorphic to the set of the signed permutations, ordered so that y . . . y n · > x . . . x n if and only if one of the two following conditions is satisfied:Either y i = x i +1 > x i = y i +1 for i, i + 1 ∈ [ n ] and y j = x j for each j (cid:54)∈ { i, i + 1 } ,or 0 < x = − y and x j = y j for all j ∈ { , , . . . , n } . In the former case, the symmetry y i = − y − i implies that y − i − = x − i > x − i − = y − i . In particular, in itsfull one-line notation, the signed permutation y has two descents: y i > y i +1 and y − i − > y − i . We say that such a pair of descents, or a single symmetric descent inpositions − type-B descent . (For more information on this realizationof the weak order on the type- B Coxeter group see [7, Section 8.1]).To motivate the definition of noncrossing arc diagrams of type B, we considerthe action y (cid:55)→ w yw on A n − where w is the longest element. (We describe w ∈ A n − below. For the general definition of length, see Section 4.3.) We writeeach element y in A n − as a permutation of {± , . . . , ± n } . In the noncrossing arcdiagram δ ( y ), we label the points − n, − n + 1 , . . . , − , , . . . , n − , n from bottomto top. We place these points so that a half-turn rotation through the center ofthe diagram maps each point i to the point − i . We call this rotation the centralsymmetry . The longest element w in this copy of A n − is the permutation( − n ) · · · ( − · · · n . Conjugation by w acts by negating all of the entries of thefull one-line notation of y and reversing its order. Thus, y is fixed by the actionof w if and only if y is a signed permutation. On the level of noncrossing arcdiagrams, the action of w coincides with the central symmetry.A centrally symmetric noncrossing arc diagram is a noncrossing arc di-agram on the points − n, . . . , − , , . . . , n that is fixed by the central symmetry.The map δ restricts to a bijection from signed permutations to centrally symmetricnoncrossing arc diagrams. We use the term centrally symmetric arc to describeeither an arc that is fixed by the central symmetry or a pair of arcs that forman orbit under the symmetry. For each k , the map δ restricts further to a bijec-tion between signed permutations with k type-B descents and centrally symmetricnoncrossing arc diagrams with k centrally symmetric arcs. Since each signed per-mutation has at most one symmetric descent in the positions − B n , are s = ( − s i = ( − i − − i )( i i +1) for i = 1 , . . . , n − {± , . . . , ± n } . A symmetric Cox-eter element of A n − is a Coxeter element that is fixed by the automorphism y (cid:55)→ w yw . Equivalently, the Coxeter element can be written as a product of somepermutation of the elements s , . . . , s n − defined above. This product in A n − canbe interpreted as a Coxeter element of B n , which we denote by ˜ c . A Coxeter ele-ment is symmetric if and only if it corresponds to a barring of (cid:8) ± , . . . , ± ( n − (cid:9) with the property that i is overbarred if and only if − i is underbarred. Thus, asigned permutation avoids the pattern 231 if and only if it also avoids the pat-tern 312 (in its full one-line notation). The signed permutations avoiding 231 (andequivalently 312) in their full notation are exactly the ˜ c -sortable elements by [33,Theorem 7.5]. Comparing with the description of c -sortable permutations followingProposition 3.1, we obtain the following proposition. Proposition 3.15.
Suppose c is a symmetric Coxeter element of A n − and sup-pose ˜ c is the corresponding Coxeter element of B n . A signed permutation is ˜ c -sortable in B n if and only if it is c -sortable as an element of A n − . The analogous result holds for ˜ c -bisortable elements. OXETER-BICATALAN COMBINATORICS 27
Proposition 3.16.
Suppose c is a symmetric Coxeter element of A n − and sup-pose ˜ c is the corresponding Coxeter element of B n . A signed permutation is ˜ c -bisortable in B n if and only if it is c -bisortable as an element of A n − .Proof. Suppose w is a signed permutation. If w is ˜ c -bisortable, then Proposi-tion 2.15 says that w = u ∨ v for some ˜ c -sortable signed permutation u and some˜ c − -sortable signed permutation v . Proposition 3.15 says that, as elements of A n − , u is a c -sortable permutation and v is a c − -sortable permutation. It iswell-known that the weak order on B n is a sublattice of the weak order on A n − .Indeed, in any finite Coxeter group, the map y (cid:55)→ w yw is a rank-preserving lat-tice automorphism. For any lattice automorphism, the set of fixed points of theautomorphism is a sublattice. Thus, the join u ∨ v is the same in A n − as in B n ,and Proposition 2.15 implies that w is c -bisortable.On the other hand, if w is c -bisortable as an element of A n − , then as inProposition 2.15, we can write w as u ∨ v , where u is the c -sortable permutation π c ↓ ( w ) and v is the c − -sortable permutation π c − ↓ ( w ). Since conjugation by w isa lattice automorphism fixing w , we obtain w = ( w uw ) ∨ ( w vw ). But w uw is c -sortable and below w , so w uw ≤ u . Since conjugation by w is order preserving,we conclude that w uw = u . Similarly w vw = v . Thus, by Proposition 3.15, u is ˜ c -sortable and v is ˜ c − -sortable in B n . Since the weak order on B n is a sublatticeof the weak order on A n − , Proposition 2.15 says that w is ˜ c -bisortable. (cid:3) A bipartite Coxeter element ˜ c of B n is a symmetric bipartite Coxeter element of A n − , so combining Propositions 3.5 and 3.16, we immediately obtain the followingproposition. Proposition 3.17.
For W = B n and ˜ c a bipartite Coxeter element, the map δ restricts to a bijection from ˜ c -bisortable signed permutations with k descents to cen-trally symmetric alternating arc diagrams on n points with k centrally symmetricalternating arcs. Thus, to count the bipartite c -bisortable elements in B n , it remains only to countcentrally symmetric alternating arc diagrams. The points in the noncrossing arcdiagram for a permutation in S n are labeled 1 , . . . , n from bottom to top. If weinstead label the points − n, . . . , − , , . . . , n from bottom to top, we can interpretthe map π as returning an ordered pair of subsets of {− n, . . . , − , , . . . n − } .Define π B to be the map on centrally symmetric alternating arc diagrams with 2 n vertices that first does the map π to obtain ( S, T ) and then ignores T and outputsonly S . The following theorem shows that the number of centrally symmetricalternating arc diagrams with k centrally symmetric arcs is (cid:0) n − k (cid:1) + (cid:0) n − k − (cid:1) = (cid:0) n k (cid:1) ,proving Theorem 1.5 for type B n . Theorem 3.18.
For each k , the map π B restricts to a bijection from centrallysymmetric alternating arc diagrams with k centrally symmetric arcs to subsets of {− n, . . . , − , , . . . n − } of size k or k − .Proof. We first show that π B is a bijection from centrally symmetric alternatingarc diagrams to subsets of {− n, . . . , − , , . . . n − } . Given S ⊆ {± , . . . , ± n } , wewrite − S − {− i − i ∈ S } , where we interpret 1 − − − S − {± , . . . , ± n } . We show that π B is a bijectionby showing that an alternating diagram Σ is centrally symmetric if and only if π (Σ) = ( S, − S −
1) for some S . The terms “right-even alternating” and “left-even alternating” should be under-stood in terms of the labeling of points as 1 , . . . , n . These terms become prob-lematic when we label points as − n, . . . , − , , . . . , n . (For example, whether aright-even alternating arc passes left or right of the point labeled i depends on thesign of i , the parity of i , and the parity of n .) Without worrying about these details,we make two easy observations: First, an alternating arc is right-even alternating ifand only if its image under the central symmetry is right-even alternating. Second,the central symmetry swaps top with bottom endpoints and positive with negativeendpoints. These observations immediately imply that π maps centrally symmetricalternating arc diagrams to pairs of the form ( S, − S − π maps an alternating arcdiagram Σ to ( S, T ) and Σ (cid:48) is the image of Σ under the central symmetry, then π maps Σ (cid:48) to ( − T + 1 , − S − − T + 1 is the set {− i + 1 : i ∈ T } , wherewe interpret − π maps Σ to ( S, − S − π also maps Σ (cid:48) to ( S, − S − π is a bijection, weconclude that in this case Σ must be centrally symmetric. We have shown that Σis centrally symmetric if and only if π (Σ) is of the form ( S, − S − π B is a bijection.It is now immediate that π B maps a centrally symmetric alternating arc diagramwith k centrally symmetric arcs to a (2 k − k -element set if all of the arcs in thediagram come in symmetric pairs. (Recall that the diagram has at most one arcfixed by the central symmetry.) (cid:3) Simpliciality of the bipartite biCambrian fan in types A and B.
Wenow prove Theorem 2.12, which states that the bipartite biCambrian fan is simpli-cial in types A and B. The proof of the type-A case of Theorem 2.12 proceeds bycombining results of [13] and [37].Some collections of noncrossing arc diagrams (including, we will see, the alter-nating arc diagrams), correspond to lattice quotients of the weak order. Morespecifically, a collection of noncrossing arc diagrams may be the image, under δ , ofthe bottom elements of congruence classes of some congruence. To describe whenand how such a situation arises, we need the notion of a subarc. For i < j and i (cid:48) < j (cid:48) , an arc α connecting i to j is a subarc of an arc α (cid:48) connecting i (cid:48) to j (cid:48) if i (cid:48) ≤ i and j (cid:48) ≥ j and if α and α (cid:48) pass to the same side of every point between i and j . It follows from [37, Theorem 4.1] and [37, Theorem 4.4] that a subset D ofthe noncrossing arc diagrams on n + 1 points is the image, under δ , of the set ofbottom elements for some congruence Θ if and only if all of the following conditionshold.(i) There exists a set U of arcs such that a noncrossing arc diagram Σ is in D ifand only if all arcs in Σ are in U .(ii) Any subarc of an arc in U is itself also in U .We will call U the set of unremoved arcs of the congruence Θ. If C is any set ofarcs and U is the maximal set such that U ∩ C = ∅ and condition (ii) above holds,then we say that the congruence Θ is generated by removing the arcs C .An element j of a finite lattice L is join-irreducible if it covers exactly oneelement j ∗ . A lattice congruence on L contracts a join-irreducible element j ifthe congruence has j ≡ j ∗ . A congruence is uniquely determined by the set of OXETER-BICATALAN COMBINATORICS 29 join-irreducible elements it contracts. The join-irreducible elements of the weakorder on A n are the permutations in S n +1 with exactly one descent. In particular,the map δ restricts to a bijection between join-irreducible elements in S n +1 andnoncrossing arc diagrams with exactly one arc. (We will think of this restrictionas mapping join-irreducible elements to arcs, rather than to singletons of arcs.)Under this bijection, the join-irreducible elements not contracted by a congruenceΘ correspond to the arcs in U , where U is the set of unremoved arcs of Θ. Thecongruence is generated by contracting a set J of join-irreducible elements ifand only if it is generated by removing the arcs δ ( J ).We call j a double join-irreducible element if it is join-irreducible and if theunique element j ∗ covered by j is either the bottom element of the lattice or is itselfjoin-irreducible. The following is a result of [13]. Theorem 3.19.
Suppose Θ is a lattice congruence on the weak order on A n . Thenthe following three conditions are equivalent. (i) The undirected Hasse diagram of the quotient lattice A n / Θ is a regular graph. (ii) F Θ ( A n ) is a simplicial fan. (iii) Θ is generated by contracting a set of double join-irreducible elements. We now apply these considerations to alternating arc diagrams. First, it isapparent that the set of alternating arc diagrams is the image of δ restricted to theset of bottom elements of a congruence. (Indeed, this is the bipartite c -biCambriancongruence.) It is also apparent that the congruence is generated by removing thearcs that connect i to i + 3 and that do not alternate. (That is they pass to thesame side of i + 1 and i + 2.) Applying the inverse of δ , we see that the congruenceis generated by contracting the join-irreducible elements1 · · · ( i − i + 1)( i + 2)( i + 3) i ( i + 4) · · · ( n + 1)and 1 · · · ( i − i + 3) i ( i + 1)( i + 2)( i + 4) · · · ( n + 1)for i = 1 , . . . , n −
2. These are both double join-irrreducible elements, and thusTheorem 3.19 implies the type-A case of Theorem 2.12.We now move to the type-B case of Theorem 2.12. Just as in type-A, thereis a correspondence between congruences on the weak order and certain sets of(centrally symmetric) noncrossing arc diagrams. However, there is currently noanalogue to Theorem 3.19 in type B. Therefore, instead of arguing the type-B caseas we argued the type-A case, we will use a folding argument to show that thetype-A case implies the type-B case.Say a lattice congruence of the weak order on A n − is symmetric underconjugation by w if for all x, y ∈ A n − we have x ≡ y modulo Θ if and only if w xw ≡ w yw modulo Θ. Proposition 3.20. If Θ is a lattice congruence of the weak order on A n − thatis symmetric under conjugation by w , then its restriction to the sublattice B n is acongruence Θ (cid:48) . An element of B n is the bottom element of its Θ (cid:48) -class if and onlyif it is the bottom element of its Θ -class.Proof. It is also a well-known and easy fact that the restriction of a lattice con-gruence to any sublattice is a congruence on the sublattice, and the first assertionof the proposition follows. One implication in the second assertion is immediate.For the other implication, suppose x ∈ B n is the bottom element of its Θ (cid:48) -class and let y = π Θ ↓ ( x ), so that in particular x ≡ y modulo Θ. Then because Θ issymmetric under conjugation by w , also x = w xw ≡ w yw modulo Θ. Since y is the bottom element of its Θ-class, y ≤ w yw . Since conjugation by w is orderpreserving, also w yw ≤ y , so y = w yw . Thus y is in the Θ (cid:48) -class of x , and weconclude that y = x , so that x is also the bottom element of its Θ-class. (cid:3) Proposition 3.21.
Suppose that Θ is a lattice congruence of the weak order on A n − and let Θ (cid:48) denote its restriction to the weak order on B n . If F Θ ( A n − ) issimplicial and Θ is symmetric under conjugation by w , then F Θ (cid:48) ( B n ) is simplicial. Before we proceed with the proof of Proposition 3.21 we define some usefulterminology. Recall that there is a linear functional λ that orients the adjacencygraph on maximal cones in F ( W ) to yield a partial order isomorphic to the weakorder on W . A facet of a maximal cone is a lower wall (with respect to λ ) ifpassing through it to an adjacent maximal cone is the same as moving down bya cover in the weak order. Upper walls are defined dually. The maximal conesof F Θ ( W ) similarly have lower and upper walls with respect to λ ; passing fromone cone to an adjacent cone through a lower wall corresponds to moving downby a cover in the lattice quotient induced by Θ. The lower walls of a maximalcone in F Θ ( W ) are the lower walls of the smallest element in the correspondingΘ-congruence class. (Recall that each maximal cone in F Θ ( W ) is the union of theset of maximal cones in F ( W ) in the same Θ-congruence class.) Dually, the upperwalls of a maximal cone in F Θ ( W ) are the upper walls of the cone correspondingto the largest element in the Θ-congruence class. Proof of Proposition 3.21.
We begin by considering type A n − in the usual geo-metric representation in R n . However, to prepare for the type-B construction,we index the standard unit basis vectors of R n as − n, . . . , − , , . . . , n . In thisrepresentation, there is a reflecting hyperplane H ji , with normal vector e j − e i ,for each i < j with i, j ∈ {± , . . . ± n } . The maximal cone corresponding to thepermutation x − n · · · x − x · · · x n has a lower (respectively upper) wall contained in H ji if and only if there exists r ∈ {− n, . . . , − , , . . . n − } such that x r = j and x r +1 = i (respectively, x r +1 = j and x r = i ). As the price for our choice of indices,when r = −
1, we must interpret r + 1 here to mean 1.Recall that the signed permutations of B n are exactly the permutations in A n − that are fixed under conjugation by w and that the restriction of weak orderto these w -fixed permutations is weak order on B n . As an abuse of terminol-ogy, the linear map on R n that sends each vector ( v − n , . . . , v − , v , . . . , v n ) to − ( v n , . . . , v , v − , . . . , v − n ) will be called the conjugation action of w on R n . Let L be the linear subspace of R n consisting of vectors fixed by this action. Theseare the vectors with v i = − v − i for all i . A permutation in A n − is fixed underconjugation by w if and only if its corresponding cone in F ( A n − ) intersects L in its relative interior, in which case the cone is also fixed under conjugation by w . Thus, we obtain F ( B n ) as the fan induced on L by F ( A n − ), and the weakorder on B n arises from that induced fan, ordered by the same linear functional λ as F ( A n − ). Moreover, F Θ (cid:48) ( B n ) is the fan induced on L by F Θ ( A n − ).Almost all of the lower walls of a w -fixed maximal cone C in F Θ ( A n − ) intersect L in pairs. Specifically, Proposition 3.20 implies that any such cone is associatedto a signed permutation x = x − n · · · x − x · · · x n that is the bottom element of itsΘ-class. A descent x − x of x contributes a single lower wall to C , and thus a single OXETER-BICATALAN COMBINATORICS 31 lower wall to C ∩ L . We will say that such a lower wall is centrally symmetric. Allother descents of x come in symmetric pairs x − i − x − i and x i x i +1 , contributing twolower walls to C . However, these two walls have the same intersection with L andthus contribute only one lower wall to C ∩ L . Similar dual statements hold for theupper walls. Most importantly, among all of the walls of C ∩ L , there are at mosttwo that are centrally symmetric: at most one among the set of lower walls, and atmost one among set of upper walls.Since F Θ ( A n − ) is simplicial, C has an odd number of walls. In particular,this implies that among all of the walls for C , there is exactly one that is centrallysymmetric wall. Suppose that this wall is a lower wall. Then, C has an oddnumber of lower walls, say 2 k −
1, and their intersection with L yields k lower wallsfor the corresponding cone C ∩ L in F Θ (cid:48) ( B n ). Since F Θ ( A n − ) is simplicial, thereare 2 n − k upper walls, which intersect L in pairs, to form n − k upper walls in F Θ (cid:48) ( B n ). Thus the cone associated to C in F Θ (cid:48) ( B n ) has a total of n walls. Thesame argument (switching lower walls with upper walls) shows that if the centrallysymmetric wall is an upper wall, the cone associated to C in F Θ (cid:48) ( B n ) has n walls.We conclude that F Θ (cid:48) ( B n ) is simplicial. (cid:3) Proof of the type-B case of Theorem 2.12.
Let c be a bipartite Coxeter element in A n − and let ˜ c be the same element, thought of as a Coxeter element of B n . Recallthat ˜ c is also bipartite.Using the bipartite case of Proposition 3.1 (with n replaced by 2 n − x ≡ y modulo Θ c if and only if w xw ≡ w yw modulo Θ c . It followsthat the c -biCambrian congruence is symmetric under conjugation by w . Sincea congruence is uniquely determined by the set of bottom elements of its classes,Proposition 3.16 implies that the restriction of the c -biCambrian congruence to B n is the ˜ c -biCambrian congruence. Thus the type-B case of the theorem follows fromProposition 3.21 and the type-A case of the theorem. (cid:3) Double-positive Catalan numbers and biCatalan numbers
For each finite Coxeter group W , the positive W -Catalan number Cat + ( W )is defined from the W -Catalan number Cat( W ) by inclusion-exclusion. In thissection, we review the definition of the positive W -Catalan number and definethe double-positive W -Catalan number Cat ++ ( W ) from the positive W -Catalannumber by inclusion-exclusion. We then prove Theorems 1.2 and 1.3 by showinghow to count both antichains in the doubled root poset and bipartite c -bisortableelements by the same formula involving double-positive Catalan numbers. Recallthat these two theorems in particular establish that the terms “biCatalan number”and “biNarayana number” make sense. As we prove these theorems, we obtainas a by-product a formula for the W -biCatalan numbers in terms of the double-positive Catalan numbers of parabolic subgroups of W . This formula leads toa recursion for the W -biCatalan numbers. Using a similar recursion for the W -Catalan numbers and a few other enumerative facts, we solve that recursion forbiCat( D n ) to complete the proof of Theorem 1.4. The recursions discussed here allhave Narayana q -analogues, but we are not at this time able to solve the recursionto find a formula for biCat( D n ; q ). See Section 4.9 for a brief discussion of thetype-D biNarayana numbers. The W -Catalan number has a uniform formula Cat( W ) = (cid:81) ni =1 h + e i +1 e i +1 , butthis formula will not play a large role in this paper. Similarly, the positive W -Catalan number is Cat + ( W ) = (cid:81) ni =1 h + e i − e i +1 , but we will give, and use, the moresimple-minded inclusion-exclusion definition of Cat + ( W ). The positive W -Catalanand positive W -Narayana numbers have interpretations in each setting of Coxeter-Catalan combinatorics. (See for example [1, 2, 4, 20, 23, 29, 34, 43].) In this paper,we give the usual interpretations in the settings of nonnesting partitions and c -sortable elements, specifically in Sections 4.2 and 4.5. We are not aware of anyuniform formulas for the double-positive W -Catalan or W -Narayana numbers; wedefine these numbers by inclusion-exclusion. Case-by-case formulas for the double-positive W -Catalan numbers are found in Theorem 4.52.The double-positive W -Narayana numbers appeared in [3] as the local h -vectorof the positive part of the cluster complex. (See Remark 4.7.) As far as we know,[3] was the first appearance of the double-positive W -Catalan/Narayana numbersand the only appearance before the current paper.4.1. Double-positivity.
We write S for the set of simple reflections generating W .Given J ⊆ S , the notation W J stands for the subgroup of W generated by J . Thesubgroup W J is called a standard parabolic subgroup of W and is a Coxetergroup in its own right with simple reflections J . In particular, each W J has aCatalan number. As usual, we define the positive W -Catalan number to be(4.1) Cat + ( W ) = (cid:88) J ⊆ S ( − | S |−| J | Cat( W J ) . As is not usual, we define the double-positive W -Catalan number to be(4.2) Cat ++ ( W ) = (cid:88) J ⊆ S ( − | S |−| J | Cat + ( W J ) . We will prove the following formula for the biCatalan numbers.
Theorem 4.1.
For any finite Coxeter group W with simple generators S , (4.3) biCat( W ) = (cid:88) | S |−| I |−| J | Cat ++ ( W I ) Cat ++ ( W J ) , where the sum is over all ordered pairs ( I, J ) of disjoint subsets of S . We can prove a refinement of Theorem 4.1 using the usual notion of positiveNarayana numbers and a notion of double-positive Narayana numbers. The posi-tive W -Narayana numbers are(4.4) Nar + k ( W ) = (cid:88) J ⊆ S ( − | S |−| J | Nar k ( W J ) . We define the double-positive W -Narayana number to be(4.5) Nar ++ k ( W ) = (cid:88) J ⊆ S ( − | S |−| J | Nar + k −| S | + | J | ( W J ) . In all of the settings where the Narayana numbers appear, it is apparent thatNar k ( W ) = 0 whenever k < k is greater than the rank of W . These definitionsestablish that Nar + k ( W ) = Nar ++ k ( W ) = 0 as well for those values of k . OXETER-BICATALAN COMBINATORICS 33
Defining Cat + ( W ; q ) = (cid:80) k Nar + k ( W ) q k and Cat ++ ( W ; q ) = (cid:80) k Nar ++ k ( W ) q k ,equations (4.4) and (4.5) correspond to(4.6) Cat + ( W ; q ) = (cid:88) J ⊆ S ( − | S |−| J | Cat( W J ; q ) . and(4.7) Cat ++ ( W ; q ) = (cid:88) J ⊆ S ( − q ) | S |−| J | Cat + ( W J ; q ) . Taking biCat( W ; q ) = (cid:80) k biNar k ( W ) q k , we will prove the following q -analog ofTheorem 4.1. Theorem 4.2.
For any finite Coxeter group W with simple generators S , (4.8) biCat( W ; q ) = (cid:88) q | M | Cat ++ ( W I ; q ) Cat ++ ( W J ; q ) , where the sum is over all ordered triples ( I, J, M ) of pairwise disjoint subsets of S . The following theorem is equivalent to Theorem 4.2.
Theorem 4.3.
For any finite Coxeter group W with simple generators S and any k , (4.9) biNar k ( W ) = (cid:88) k −| M | (cid:88) i =0 Nar ++ i ( W I ) Nar ++ k −| M |− i ( W J ) , where the outer sum is over all ordered triples ( I, J, M ) of pairwise disjoint subsetsof S . (If | M | > k , then the inner sum is interpreted to be zero.) To prove these theorems, as well as Theorems 1.2 and 1.3, we establish (inPropositions 4.8 and 4.29) that the right side of (4.8) counts antichains A in thedoubled root poset with weight q | A | and also counts bipartite c -bisortable elements v with weight q des( v ) . Once these counts are established, Theorems 1.2 and 1.3 follow,and in particular the definitions of the biCatalan and biNarayana numbers arevalidated. Also, Theorem 4.2 holds, leading immediately to Theorems 4.1 and 4.3.4.2. Counting twin nonnesting partitions.
We now recall the interpretationsof the positive Catalan and Narayana numbers and give the interpretations ofdouble-positive Catalan and Narayana numbers in the nonnesting setting. (Resultsin [4, 43] give the same interpretations, but accomplish much more, by establishingbijections and counting formulas. By contrast, here we are only making simpleassertions about inclusion-exclusion.) After giving these interpretations, we provethat the formula in Theorem 4.3 counts k -element antichains in the doubled rootposet.Since it is customary to talk about the “ W -Catalan number” rather than the “Φ-Catalan number,” we will make statements about “the root poset of W ,” when W isa crystallographic Coxeter group. This is harmless because, although the map fromcrystallographic root systems to Coxeter groups is not one-to-one, for each crys-tallographic Coxeter group, all corresponding crystallographic root systems haveisomorphic root posets. Correspondingly, when W J is a standard parabolic sub-group of W , we will say that a root or set of roots is “contained in W J ” if it iscontained in the subset of Φ forming a root system for W J . An antichain that is notcontained in any proper parabolic W J has full support, in the sense of Section 2.1.For any J ⊆ S , the number of antichains in the root poset for W that arecontained in W J is Cat( W J ). By inclusion-exclusion, we conclude that: Proposition 4.4.
The number of antichains in the root poset for W with fullsupport is Cat + ( W ) . The number of k -element antichains in the root poset for W with full support is Nar + k ( W ) . For J ⊆ S , the map A (cid:55)→ A \ { α i : i ∈ J } is a bijection from the set of antichainscontaining the simple roots { α i : i ∈ J } to the set of antichains in the root posetfor W S \ J .Using this bijection, we prove the following proposition. Proposition 4.5.
The number of antichains in the root poset for W containingno simple roots is Cat + ( W ) . The number of k -element antichains in the root posetfor W containing no simple roots is Nar + n − k ( W ) .Proof. The bijection mentioned above implies that the generating function for an-tichains containing the simple roots { α i : i ∈ J } (and possibly additional simpleroots) is q | J | Cat( W S \ J ; q ). By inclusion-exclusion, the generating function for k -element antichains containing no simple roots is (cid:80) J ⊆ S ( − q ) | S |−| J | Cat( W J ; q ). Onthe other hand, starting with (4.6), replacing q by q − , multiplying through by q | S | (i.e. q n ), and using the known symmetry q | J | Cat( W J ; q − ) = Cat( W J ; q ) of thecoefficients of Cat( W J ; q ), we obtain n (cid:88) k =0 Nar + n − k ( W ) q k = (cid:88) J ⊆ S ( − q ) | S |−| J | Cat( W J ; q ) . (cid:3) The bijection described above restricts to a bijection from the set of antichains with full support containing the simple roots { α i : i ∈ J } to the set of antichains with full support in the root poset for W S \ J . Thus, a similar inclusion-exclusionargument yields the following proposition. Proposition 4.6.
The number of antichains in the root poset for W with full sup-port containing no simple roots is Cat ++ ( W ) . The number of k -element antichainsin the root poset for W with full support containing no simple roots is Nar ++ k ( W ) . Remark 4.7.
The polynomials Cat ++ ( W ; q ) appeared in [3], where Athanasiadisand Savvidou showed that Cat ++ ( W ; q ) is the local h -vector of the positive partof the cluster complex, as we now explain. We refer to [3] for the relevant def-initions, which we will not need here. In light of [4, Theorem 1.5] and Proposi-tion 4.5, the polynomial h (∆ + (Φ) , x ) appearing in [3] is x | S | Cat + ( W ; x − ), where( W, S ) is the Coxeter system associated to Φ. Thus the assertion of [3, Proposi-tion 2.5] is that the local h -vector of the positive part of the cluster complex is (cid:80) J ⊆ S ( − | S |−| J | x | J | Cat + ( W ; x − ). But since the local h -vector is symmetric by[44, Theorem 3.3], we can replace x by x − and multiply by x | S | to show that thelocal h -vector is (cid:80) J ⊆ S ( − x ) | S |−| J | Cat + ( W ; x ) = Cat ++ ( W ; x ).We now prove the key result on antichains in the doubled root poset. Proposition 4.8.
For any finite Coxeter group W with simple generators S , thegenerating function (cid:80) A q | A | for antichains A in the doubled root poset is (4.10) (cid:88) q | M | Cat ++ ( W I ; q ) Cat ++ ( W J ; q ) , where the sum is over all ordered triples ( I, J, M ) of pairwise disjoint subsets of S . OXETER-BICATALAN COMBINATORICS 35
Proof.
In light of Proposition 4.6, the proposition amounts to the following asser-tions: First, there is a bijection from antichains A in the doubled root poset totriples ( B, C, M ) such that B and C are antichains in the root poset for W , eachcontaining no simple roots, and the sets I = supp( B ), J = supp( C ) and M are pair-wise disjoint. Second, under this bijection, | B | + | C | + | M | = | A | . Every antichain A in the doubled root poset consists of some set B of positive non-simple rootsin the top root poset, some set C of positive non-simple roots in the bottom rootposet, and some set M of simple roots. The sets I , J , and M are pairwise disjointbecause A is an antichain. The map A (cid:55)→ ( B, C, M ) is the desired bijection. (cid:3)
It will be useful to have a similar formula for antichains in the (not doubled)root poset, which are known to be counted by Cat( W ). Theorem 4.9.
For any finite Coxeter group W with simple generators S . (4.11) Cat( W ; q ) = (cid:88) q | J | Cat ++ ( W I ; q ) , where the sum is over all ordered pairs ( I, J ) of disjoint subsets of S .Proof. Every antichain A in the root poset consists of some set B of positive non-simple roots and some set C of simple roots. Writing I and J for the supports of B and C , again I and J are disjoint. By Proposition 4.6, each pair ( I, J ) of disjointsubsets of S contributes q | J | Cat ++ ( W I ; q ) to the count. (cid:3) The following is an immediate consequence of Proposition 4.6 and will also beuseful.
Proposition 4.10. If W is reducible as W × W , then (4.12) Cat ++ ( W ; q ) = Cat ++ ( W ; q ) Cat ++ ( W ; q ) . Canonical join representations and lattice congruences.
To count bi-partite c -bisortable elements, we will use a canonical factorization in the weak ordercalled the canonical join representation. In this section, we focus exclusively on thelattice-theoretic tools that we will use in the following sections to complete theproof of Theorem 4.3. Additional background specific to lattice congruences canbe found in Section 2.3.The canonical join representation is a “minimal” expression for an element as ajoin of join-irreducible elements. The construction is somewhat analogous to primefactorizations of integers. Indeed, in the divisibility poset for positive integers,where p ≤ q if and only if p | q , the canonical join representation coincides withprime factorization. For our purposes, the canonical join representation is usefulbecause of how it interacts with lattice congruences. Recall that a lattice congruenceΘ contracts a join-irreducible element j if j is equivalent modulo Θ to the uniqueelement that it covers. Each congruence Θ of a finite lattice is determined bythe set of join-irreducible elements that it contracts. In particular, we can seewhich elements of W are c -sortable or c -bisortable by looking at their canonicaljoin representations (much as we looked at the arcs in their arc diagrams in typesA and B).The canonical join representation of an element a is an expression a = (cid:87) A suchthat A is minimal in two senses, among sets joining to a . First, the join (cid:87) A is irredundant , meaning that there is no proper subset A (cid:48) ⊂ A with (cid:87) A (cid:48) = (cid:87) A .Second, A has the smallest possible elements (in terms of the partial order on L ). Specifically, a subset A of L join-refines a subset B of L if for each a ∈ A thereis an element b ∈ B such that a ≤ b . Join-refinement is a preorder on the subsetsof L that restricts to a partial order on the set of antichains. The canonical joinrepresentation of a , if it exists, is the unique minimal antichain A , in the sense ofjoin-refinement, that joins irredundantly to a . We sometimes write Can( a ) for A .The elements of A are called the canonical joinands of a . It follows immediatelythat each canonical joinand is join-irreducible.Not every finite lattice admits a canonical join representation for each of itselements. For example, in the diamond lattice M , which has five elements, three ofwhich are atoms, the largest element does not have a canonical join representation.Many interesting lattices do admit canonical join representations, including all finitedistributive lattices and, as we will see, the weak order on finite Coxeter groups.The next proposition establishes the promised connection between canonical joinrepresentations and lattice congruences. (The last assertion in the proposition alsofollows from [36, Proposition 6.3].) Proposition 4.11.
Suppose L is a finite lattice such that each element in L has acanonical join representation, and suppose that Θ is a lattice congruence on L . If j is a canonical joinand of a ∈ L and j is not contracted by Θ , then j is a canonicaljoinand of π Θ ↓ ( a ) in L . Moreover, if π Θ ↓ ( a ) = a then none of the canonical joinandsof a are contracted by Θ . The assertion that j is a canonical joinand of π Θ ↓ ( a ) in L implies also that j is acanonical joinand of π Θ ↓ ( a ) in π Θ ↓ ( L ). (Since π Θ ↓ ( L ) is a join-sublattice of L , everyjoin-representation of π Θ ↓ ( a ) in π Θ ↓ ( L ) is also a join-representation of π Θ ↓ ( a ) in L .) Proof.
Throughout the proof, we write { j , . . . j k } for Can( a ) with j = j . Recallthat the lattice quotient L/ Θ is isomorphic to the subposet of L induced by theset π Θ ↓ ( L ). Suppose j is not contracted by Θ, so that π Θ ↓ ( j ) = j . Recall that π Θ ↓ is a lattice homomorphism, so π Θ ↓ ( a ) = (cid:87) ki =1 π Θ ↓ ( j ) = j ∨ (cid:16)(cid:87) ki =2 π Θ ↓ ( j i ) (cid:17) ,(where the joins are all taken in the lattice quotient L/ Θ). Since L/ Θ is also ajoin-sublattice of L , the join in L/ Θ coincides with the join in L . Thus π Θ ↓ ( a )is equal to j ∨ (cid:16)(cid:87) ki =2 π Θ ↓ ( j i ) (cid:17) in L . Write B for the set Can( π Θ ↓ ( a )). Thus B join-refines { j } ∪ (cid:110) π Θ ↓ ( j ) , . . . , π Θ ↓ ( j k ) (cid:111) . If no element of B is less or equal to j ,then this join-refinement implies that each element of B is below some element of (cid:110) π Θ ↓ ( j ) , . . . , π Θ ↓ ( j k ) (cid:111) , so that π Θ ↓ ( a ) ≤ (cid:87) ki =2 π Θ ↓ ( j i ). Since also π Θ ↓ ( a ) is equal to j ∨ (cid:16)(cid:87) ki =2 π Θ ↓ ( j i ) (cid:17) , we see that j ≤ (cid:87) ki =2 π Θ ↓ ( j i ). Recall that π Θ ↓ ( j i ) ≤ j i for each i ,so we have j ≤ (cid:87) ki =2 j i . This contradicts the fact that (cid:87) ki =1 j i is irredundant. Weconclude that there is some j (cid:48) ∈ B with j (cid:48) ≤ j . Observe that (cid:0)(cid:87) B (cid:1) ∨ (cid:16)(cid:87) ki =2 j i (cid:17) = a because j = j ≤ π Θ ↓ ( a ) ≤ a . Thus, { j , . . . j k } join-refines B ∪ { j , . . . j k } . Since j is incomparable to each j i , there is some j (cid:48)(cid:48) ∈ B such that j ≤ j (cid:48)(cid:48) . But B is anantichain, so j (cid:48) = j (cid:48)(cid:48) = j , and thus j ∈ B as desired.Now suppose that π Θ ↓ ( a ) = a . Then a = (cid:87) ni =1 π Θ ↓ ( j i ), so { j , . . . j k } join-refines (cid:110) π Θ ↓ ( j ) , . . . , π Θ ↓ ( j k ) (cid:111) . Thus, for each j i , there is some j m with j i ≤ π Θ ↓ ( j m ). But OXETER-BICATALAN COMBINATORICS 37 π Θ ↓ ( j m ) ≤ j m , and since { j , . . . j k } is an antichain, we have j i = j m , and thus also j i = π Θ ↓ ( j i ). (cid:3) We will use the following easy proposition, which appears as [37, Proposition 2.2].
Proposition 4.12.
Suppose L is a finite lattice and J ⊂ L . If (cid:87) J is the canonicaljoin representation of some element of L and if J (cid:48) ⊆ J , then (cid:87) J (cid:48) is the canonicaljoin representation of some element of L . Next we consider canonical join representations in the weak order. Before webegin, we briefly review some relevant terminology. For each w ∈ W , the length of w , denoted l ( w ), is the number of letters in a reduced (that is, a shortest possible)word for w in the alphabet S . The covers in the (right) weak order on W are w · > ws whenever w ∈ W and s ∈ S have l ( ws ) < l ( w ). In this case, the simple generator s is a descent of w . Let T denote the set of reflections in W . An inversion of w is a reflection t such that l ( tw ) < l ( w ). We denote the set of inversions of w by inv( w ). A cover reflection of w is an inversion t of w such that tw = ws forsome s ∈ S . Thus, the cover reflections of w are in bijection with the descents of w . We write cov( w ) for the set of cover reflections of w . The following propositionis quoted from [40, Theorem 8.1]. Proposition 4.13.
Fix a finite Coxeter group W , and an element w ∈ W . Thecanonical join representation of w exists and is equal to (cid:87) j t where t ranges overthe set of cover reflections of w , and j t is the unique smallest element below w thathas t as an inversion. In particular, w has des( w ) many canonical joinands. Recall that the support of w , written supp( w ), is the set of simple reflectionsappearing in a reduced word for w , and is independent of the choice of reducedword for w . The following lemma is an immediate consequence of the fact thatevery standard parabolic subgroup W J is a lower interval in the weak order on W . Lemma 4.14.
For each w ∈ W , the support of w equals (cid:83) j ∈ Can( w ) supp( j ) . For each element w and standard parabolic subgroup W J , there is a uniquelargest element below w that belongs to W J . We write w J for this element and π J ↓ for the map that sends w to w J . In [31, Corollary 6.10], it was shown that thefibers of π J ↓ constitute a lattice congruence of the weak order. We write Θ J for thiscongruence. Since π J ↓ sends each element to the bottom if its fiber, it is a latticehomomorphism from W to π J ↓ ( W ), which equals W J . Lemma 4.15.
Suppose that A and A are antichains with disjoint support suchthat (cid:87) A and (cid:87) A are both canonical join representations in the weak order on W .Then (cid:87) ( A ∪ A ) is a canonical join representation.Proof. We write A for A ∪ A . First we show that (cid:87) A is irredundant. By way ofcontradiction, assume that there is some j ∈ A such that (cid:87) A = (cid:87) ( A \ { j } ). Wemay as well take j ∈ A . We write J for the support of A . Since the support ofeach join-irreducible element j (cid:48) in A is disjoint from J , and since support decreasesweakly in the weak order, we conclude that π J ↓ ( j (cid:48) ) is the identity element. Since π J ↓ is a lattice homomorphism, π J ↓ ( (cid:87) A ) = (cid:87) A and π J ↓ ( (cid:87) ( A \{ j } )) = (cid:87) ( A \{ j } ). Weconclude that (cid:87) A = (cid:87) ( A \ { j } ), contradicting the fact that (cid:87) A is a canonicaljoin representation. Next we show that Can( (cid:87) A ) is contained in A . Assume that j (cid:48)(cid:48) is a canonicaljoinand of (cid:87) A . There is some j ∈ A such that j (cid:48)(cid:48) ≤ j . Assume that j ∈ A , so thatsupp( j (cid:48)(cid:48) ) ⊂ J . Thus, π J ↓ ( j (cid:48)(cid:48) ) = j (cid:48)(cid:48) . Proposition 4.11 says j (cid:48)(cid:48) is a canonical joinandof π J ↓ ( (cid:87) A ) = (cid:87) A . Because A is an antichain, j (cid:48)(cid:48) = j . Since (cid:87) A is irredundant,and A contains Can( (cid:87) A ), we conclude that A is equal to Can( (cid:87) A ). (cid:3) Observe that if s ∈ S is a cover reflection of w then Proposition 4.13 implies that s is also a canonical joinand of w because simple reflections are atoms in the weakorder. We immediately obtain the following useful fact. Lemma 4.16.
Each w ∈ W has Can( w ) ∩ S = cov( w ) ∩ S . In much of what follows, for s ∈ S , we will use the abbreviation (cid:104) s (cid:105) to mean S \ { s } . It is known (see for example [35, Lemma 2.8]) that if w ∈ W (cid:104) s (cid:105) , thencov( w ∨ s ) = cov( w ) ∪ { s } . We close this section with a lemma extends this state-ment to canonical join representations. Lemma 4.17. If w ∈ W (cid:104) s (cid:105) , then Can( w ∨ s ) = Can( w ) ∪ { s } .Proof. Since support is weakly decreasing in the weak order, each j ∈ Can( w ) hassupport contained in (cid:104) s (cid:105) . Lemma 4.15 says that (cid:87) (cid:0) Can( w ) ∪ { s } (cid:1) is a canonicaljoin representation. (cid:3) Canonical join representations of c -bisortable elements. In this sectionwe focus on canonical join representations of c -sortable elements and c -bisortableelements. Our goal is to prove the following result: Proposition 4.18.
Fix a bipartite c -bisortable element w and the correspondingtwin ( c, c − ) -sortable elements ( u, v ) = ( π c ↓ ( w ) , π c − ↓ ( w )) . Then (1) Can( w ) ∩ S = Can( u ) ∩ Can( v )(2) Can( w ) is the disjoint union (Can( u ) \ S ) (cid:93) (Can( v ) \ S ) (cid:93) (Can( w ) ∩ S )(3) The sets supp(Can( u ) \ S ) , supp(Can( v ) \ S ) and Can( w ) ∩ S are pairwisedisjoint. We begin with an easy application of Proposition 4.11 (the first item below canalso be found as [40, Proposition 8.2]).
Proposition 4.19.
For any Coxeter element c and w ∈ W :(1) w is c -sortable if and only if each of its canonical joinands is c -sortable.(2) w is c -bisortable if and only if each of its canonical joinands is either c - or c − -sortable.Proof. The first assertion follows immediately from Proposition 4.11. Recall thenotation Θ c for the c -Cambrian congruence and write Θ for the c -biCambrian con-gruence. Since Θ = Θ c ∧ Θ c − , a join-irreducible element in W is contracted by Θif and only if it is contracted by Θ c and by Θ c − . The second assertion follows. (cid:3) Recall from Section 2.4 that a simple reflection s is initial in a Coxeter element c if there is a reduced word a . . . a n for c with a = s . Similarly s is final in c ifthere is a reduced word a . . . a n for c with a n = s . In much of what follows, thekey property of a bipartite Coxeter element is that every s ∈ S is either initial orfinal in c .The following lemma is the combination of [40, Propositions 3.13, 5.3, and 5.4].Recall that v (cid:104) s (cid:105) is the largest element in W below v that belongs to W (cid:104) s (cid:105) . OXETER-BICATALAN COMBINATORICS 39
Lemma 4.20.
Fix a c -sortable element v in W and a simple reflection s ∈ S .(1) If s is final in c and v ≥ s , then v (cid:104) s (cid:105) is cs -sortable and v = s ∨ v (cid:104) s (cid:105) .(2) If s be initial in c and s ∈ cov( v ) , then v (cid:104) s (cid:105) is sc -sortable and v = s ∨ v (cid:104) s (cid:105) . Observe that if v satisfies the conditions of either item in Lemma 4.20, then byLemma 4.17, Can( v ) = { s } ∪ Can( v (cid:104) s (cid:105) ). The following two lemmas are a straight-forward application of Lemma 4.20. Lemma 4.22 is a restatement of Remark 2.7,for the special case where c is bipartite, and, for this special case, we give a simplerproof. Lemma 4.21. If j is a c -sortable join-irreducible element and s is final in c with j ≥ s , then j = s .Proof. The first assertion of Lemma 4.20 says that j = s ∨ j (cid:104) s (cid:105) . Since j is join-irreducible and not equal to j (cid:104) s (cid:105) , we conclude that j = s . (cid:3) Lemma 4.22. If c is a bipartite Coxeter element and j is a join-irreducible elementthat is both c -sortable and c − -sortable, then j is a simple reflection.Proof. Because j is join-irreducible, it is not the identity, so there is some s ∈ S such that j ≥ s . Since c is bipartite, we can assume without loss of generality that s is final in c . (If not, then replace c with c − .) Thus j = s by Lemma 4.21. (cid:3) Putting together Lemma 4.21 and Lemma 4.22, we obtain an explicit descriptionof π c − ↓ ( j ), for bipartite c -sortable join-irreducible elements. Lemma 4.23.
Suppose that c is a bipartite Coxeter element and j is a c -sortablejoin-irreducible element. Let S (cid:48) denote the set of simple reflections s such that j ≥ s . Then π c − ↓ ( j ) = (cid:87) S (cid:48) , which equals (cid:81) S (cid:48) , the product in W . Moreover, thisjoin is a canonical join representation.Proof. The statement of the lemma is obvious if j is a simple reflection, so weassume that j is not simple. Thus, Lemma 4.22 implies that j is not c − -sortable,so π c − ↓ ( j ) is strictly less than j .If any s ∈ S (cid:48) is final in c , then Lemma 4.21 says that j = s , contradicting ourassumption. Thus, since c is bipartite, each s ∈ S (cid:48) is initial. In particular, the el-ements of S (cid:48) pairwise commute, so that the notation (cid:81) S (cid:48) makes sense and equals (cid:87) S (cid:48) . Moreover, since (cid:87) S (cid:48) is an irredundant join of atoms, it is a canonical joinrepresentation. Since each simple reflection is both c - and c − -sortable, Proposi-tion 4.19 says that this element is c − -sortable. We conclude that π c − ↓ ( j ) ≥ (cid:87) S (cid:48) .Suppose that j (cid:48) is a canonical joinand of π c − ↓ ( j ). There is some simple reflection s such that j (cid:48) ≥ s . Since also j (cid:48) ≤ π c − ↓ ( j ) ≤ j , we conclude that s ∈ S (cid:48) . Everyelement of S (cid:48) is initial in c and thus final in c − , so again by Lemma 4.21, j (cid:48) = s .We conclude that Can( π c − ↓ ( j )) ⊆ S (cid:48) . Thus π c − ↓ ( j ) = (cid:87) S (cid:48) . (cid:3) Recall that Lemma 4.15 says that if j and j (cid:48) are join-irreducible elements withdisjoint support, then j ∨ j (cid:48) is canonical. In Lemma 4.25 below, we prove that when j is bipartite c -sortable and j (cid:48) is bipartite c − -sortable, the converse is also true.We begin with the case when j (cid:48) is a simple reflection. Lemma 4.24.
Given a bipartite Coxeter element c , a c -sortable join-irreducibleelement j and a simple reflection s ∈ supp( j ) , there exists no element w ∈ W withboth s and j in Can( w ) . Proof.
In light of Proposition 4.12, to prove this proposition, it is enough to showthat no element can have s ∨ j as its canonical join representation. Suppose tothe contrary that there is an element v with canonical join representation s ∨ j .By Proposition 4.19, v is c -sortable. Also s ∨ j is irredundant, so j and s areincomparable. Since c is bipartite, s is either initial or final in c , so Lemma 4.20says that v = s ∨ v (cid:104) s (cid:105) . Since v = s ∨ j is a canonical join representation, we see that j ≤ v (cid:104) s (cid:105) , contradicting the hypothesis that s is in the support of j . (cid:3) Lemma 4.25.
Fix a bipartite Coxeter element c in W . Suppose that j is a c -sortable join-irreducible element and that j (cid:48) is a c − -sortable join-irreducible ele-ment. Suppose that j ∨ j (cid:48) is a canonical join representation for some element of W .Then j and j (cid:48) have disjoint support.Proof. Suppose that s ∈ supp( j ) ∩ supp( j (cid:48) ), and assume without loss of generalitythat s is initial in c . It is immediate from the definition of c -sortable elements that s ≤ j . (See for example [40, Proposition 2.29].) Since s is a c − -sortable element,also s ≤ π c − ↓ ( j ∨ j (cid:48) ). By Lemma 4.20(1) and Lemma 4.17, s is a canonical joinandof π c − ↓ ( j ∨ j (cid:48) ). But also Proposition 4.11 says that j (cid:48) is a canonical joinand of π c − ↓ ( j ∨ j (cid:48) ). We have reached a contradiction to Lemma 4.24, and we concludethat supp( j ) ∩ supp( j (cid:48) ) = ∅ . (cid:3) Finally, we prove Proposition 4.18.
Proof of Proposition 4.18.
Lemma 4.22 implies that Can( w ) is the disjoint union(Can( w ) ∩ S ) (cid:93) J + (cid:93) J − such that J + is the set of c -sortable join-irreducible el-ements in Can( w ) \ S and J − is the set of c − -sortable join-irreducible elementsin Can( w ) \ S . Moreover, by Lemma 4.25, these sets have pairwise disjoint sup-port. For each j ∈ J − , write S (cid:48) j for the set of simple reflections s such that s ≤ j , and S (cid:48) = (cid:83) S (cid:48) j , where the union ranges over all j ∈ J − . Lemma 4.23says that π c ↓ ( j ) = (cid:87) S (cid:48) j . Since π c ↓ is a join-homomorphism, π c ↓ ( (cid:87) J − ) = (cid:87) S (cid:48) .Thus, applying the map π c ↓ to the join (cid:87) [(Can( w ) ∩ S ) (cid:93) J + (cid:93) J − ], we see that (cid:87) [(Can( w ) ∩ S ) (cid:93) J + (cid:93) S (cid:48) ] is a join representation of u . Since S (cid:48) is contained inthe support of J − , the sets Can( w ) ∩ S , J + , and S (cid:48) also have pairwise disjoint sup-port. Proposition 4.12 says that both (cid:87) Can( w ) ∩ S and (cid:87) J + are canonical joinrepresentations. Since (cid:87) S (cid:48) is an irredundant join of atoms, it is also a canonicaljoin representation. Thus, by Lemma 4.15, (cid:87) [(Can( w ) ∩ S ) (cid:93) J + (cid:93) S (cid:48) ] is the canonicaljoin representation of u . The symmetric argument gives the canonical join repre-sentation of v . We conclude that Can( w ) ∩ S = Can( u ) ∩ Can( v ), J + = Can( u ) \ S ,and J − = Can( v ) \ S . The proposition follows. (cid:3) Counting bipartite c -bisortable elements. In this section, we prove thatthe formulas in Theorem 4.3 counts bipartite c -bisortable elements, thus completingthe proofs of Theorems 1.2, 1.3, 4.1, 4.2 and 4.3. We begin by interpreting thedouble-positive Catalan and Narayana numbers in the c -sortable setting. We define positive c -sortable elements to be the set of c -sortable elements not containedin any standard parabolic subgroup of W . Equivalently, these are the c -sortableelements whose support is not contained in any proper subset of S . As the namesuggests, positive c -sortable elements are counted by the positive Catalan numbers.The following analogue of Proposition 4.4 is the combination of [34, Corollary 9.2]and [34, Corollary 9.3]. OXETER-BICATALAN COMBINATORICS 41
Proposition 4.26.
For any Coxeter element c of W , the number of positive c -sortable elements in W is Cat + ( W ) . The number positive c -sortable elements with k descents is Nar + k ( W ) . We define clever c -sortable elements to be c -sortable elements which have nosimple canonical joinands. We continue to let (cid:104) s (cid:105) stand for S \ { s } . To count clever c -sortable elements we will use Lemma 4.20 to define a map from c -sortable elements v with simple cover reflection s to c (cid:48) -sortable elements in the standard parabolicsubgroup W (cid:104) s (cid:105) , where c (cid:48) is the restriction of c to W (cid:104) s (cid:105) . Our next task is to showthat, for bipartite c , clever c -sortable elements are analogous, enumeratively, toantichains in the root poset having no simple roots: Proposition 4.27.
Fix a bipartite Coxeter element c of W .(1) The number of clever c -sortable elements is Cat + ( W ) .(2) The number of positive, clever c -sortable elements is Cat ++ ( W ) .(3) The number of positive, clever c -sortable elements with exactly k descentsis Nar ++ k ( W ) . We emphasize that while Proposition 4.26 holds for arbitrary c , Proposition 4.27holds only for bipartite c . The proof of Proposition 4.27 will use inclusion-exclusionand the following technical lemma. Lemma 4.28.
For bipartite c and J ⊆ S , let c (cid:48) be the restriction of c to W S \ J .(1) The map π S \ J ↓ : v (cid:55)→ v S \ J is a bijection from c -sortable elements of W with J ⊆ Can( v ) to c (cid:48) -sortable elements of W S \ J . Also, Can( v S \ J ) = Can( v ) \ J .(2) The map restricts to a bijection from positive c -sortable elements of W with J ⊆ Can( v ) to positive c (cid:48) -sortable elements of W S \ J .(3) The map restricts further to a bijection from positive c -sortable elementsof W with J ⊆ Can( v ) and with exactly k descents to positive c (cid:48) -sortableelements of W S \ J with exactly k − | J | descents.Proof. Suppose that v is c -sortable, and J ⊆ Can( v ). Lemma 4.24 says that thesupport of each canonical joinand j in Can( v ) \ J is contained in S \ J . (Lemma 4.24applies to the non-simple elements of Can( v ). Clearly, each simple reflection s ∈ Can( v ) \ J is supported on S \ J .) On the one hand, π S \ J ↓ ( j ) = j for each j ∈ Can( v ) \ J . On the other hand, π S \ J ↓ ( s ) is the identity element for each s in J . Since π S \ J ↓ is a lattice homomorphism, π S \ J ↓ ( (cid:87) Can( v )) = (cid:87) [Can( v ) \ J ]. Proposition 4.11implies that (cid:87) [Can( v ) \ J ] is the canonical join representation of π S \ J ↓ ( v ) = v S \ J .Lemma 4.19 says that v S \ J is c (cid:48) -sortable.To complete the proof of the first assertion, we construct an inverse map. Sup-pose that v (cid:48) is a c (cid:48) -sortable element in W S \ J . Lemma 4.14 says that the supportof each canonical joinand j ∈ Can( v (cid:48) ) is contained in S \ J . Lemma 4.15 says thatthe join (cid:87) [Can( v (cid:48) ) ∪ J ] is a canonical join representation for some element v ∈ W .Lemma 4.19 says that v is c -sortable. We conclude that the map sending v (cid:48) to (cid:87) [Can( v (cid:48) ) ∪ J ] is a well-defined inverse.Lemma 4.14, Lemma 4.24, and the fact that Can( v S \ J ) = Can( v ) \ J imply that v is positive in W if and only if v S \ J is positive in W S \ J . The second assertionfollows. The third assertion then follows from Proposition 4.13 and the fact thatCan( v S \ J ) = Can( v ) \ J . (cid:3) Finally, we complete the proof of that bipartite c -bisortable elements are countedby the formula in Theorem 4.2. Proposition 4.29.
For any finite Coxeter group W with simple generators S , thegenerating function (cid:80) v q des( v ) for bipartite c -bisortable elements is (cid:88) q | M | Cat ++ ( W I ; q ) Cat ++ ( W J ; q ) , where the sum is over all ordered triples ( I, J, M ) of pairwise disjoint subsets of S .Proof. Similarly to the proof of Proposition 4.8, the proposition amounts to estab-lishing a bijection from bipartite c -bisortable elements w to triples ( u (cid:48) , v (cid:48) , M ) suchthat u (cid:48) is a clever c -sortable element, v (cid:48) is a clever c − -sortable element, and thesets I = supp( u (cid:48) ), J = supp( v (cid:48) ), and M are disjoint subsets of S , and then showingthat des( w ) = des( u (cid:48) ) + des( v (cid:48) ) + | M | .Given a bipartite c -bisortable element w , write ( u, v ) for the corresponding pair( π c ↓ ( w ) , π c − ↓ ( w )) of twin ( c, c − )-sortable elements. Proposition 4.18(2) says thatCan( w ) is the disjoint union (cid:0) Can( u ) \ S (cid:1) (cid:93) (cid:0) Can( v ) \ S (cid:1) (cid:93) (cid:0) Can( w ) ∩ S (cid:1) . Propo-sition 4.18(3) says that the sets I = supp(Can( u ) \ S ), J = supp(Can( v ) \ S ),and M = Can( w ) ∩ S are pairwise disjoint subsets of S . By Proposition 4.12, (cid:87) Can( u ) \ S is the canonical join representation of a positive, clever c -sortableelement u (cid:48) in W I . Similarly, (cid:87) Can( v ) \ S is the canonical join representation of apositive, clever c − -sortable element v (cid:48) in W J . Applying Proposition 4.13 severaltimes, we see that des( w ) = des( u (cid:48) ) + des( v (cid:48) ) + | M | .We will show that this map w (cid:55)→ ( u (cid:48) , v (cid:48) , M ) is a bijection by showing that themap ( u (cid:48) , v (cid:48) , M ) (cid:55)→ u (cid:48) ∨ v (cid:48) ∨ ( (cid:87) M ) is the inverse. On one hand, given w , construct( u (cid:48) , v (cid:48) , M ) as above. Then w equals (cid:87) Can( w ), which equals (cid:16)(cid:95) Can( u ) \ S (cid:17) ∨ (cid:16)(cid:95) Can( v ) \ S (cid:17) ∨ (cid:16)(cid:95) Can( w ) ∩ S (cid:17) = u (cid:48) ∨ v (cid:48) ∨ ( (cid:95) M ) . On the other hand, given a triple ( u (cid:48) , v (cid:48) , M ) satisfying the description above, set w = u (cid:48) ∨ v (cid:48) ∨ ( (cid:87) M ). Since u (cid:48) , v (cid:48) and M have pairwise disjoint support, we concludethat Can( u (cid:48) ), Can( v (cid:48) ), and M also have pairwise disjoint support. Lemma 4.15says that (cid:87) Can( u (cid:48) ) (cid:93) Can( v (cid:48) ) (cid:93) M is the canonical join representation of w . ByLemma 4.19(1), each canonical joinand of u (cid:48) is c -sortable and each canonical joinandof v (cid:48) is c − -sortable. Since each simple generator is both c - and c − -sortable, weconclude that each canonical joinand of w either either c - or c − -sortable. ByLemma 4.19(2), w is c -bisortable. Thus, the map ( u (cid:48) , v (cid:48) , M ) (cid:55)→ u (cid:48) ∨ v (cid:48) ∨ ( (cid:87) M ) isa well-defined.Lemma 4.22 says that Can( u (cid:48) ) (cid:93) M is equal to the set of c -sortable canonicaljoinands of w . Since u (cid:48) is clever, Can( u (cid:48) ) is equal to the set of c -sortable canonicaljoinands in Can( w ) \ S . Similarly, Can( v (cid:48) ) is the set of c − -sortable canonicaljoinands in Can( w ) \ S , and Can( w ) ∩ S = M . Define u = π c ↓ ( w ) and v = π c − ↓ ( w ).Proposition 4.18(2) says that Can( w ) = (Can( u ) \ S ) (cid:93) (Can( v ) \ S ) (cid:93) (Can( w ) ∩ S ).Comparing this to the expression Can( w ) = Can( u (cid:48) ) (cid:93) Can( v (cid:48) ) (cid:93) M , we see thatCan( u ) \ S = Can( u (cid:48) ), that Can( v ) \ S = Can( v (cid:48) ), and that Can( w ) ∩ S = M . Thusthe map described above takes w back to ( u (cid:48) , v (cid:48) , M ). (cid:3) Remark 4.30.
The proof given here that twin nonnesting partitions are in bijectionwith bipartite c -bisortable elements would be uniform if there were a uniform proofconnecting c -sortable elements and nonnesting partitions. The opposite is true OXETER-BICATALAN COMBINATORICS 43 as well: Suppose one proved uniformly that a given map φ is a bijection fromantichains in the doubled root poset to bipartite c -bisortable elements and alsothat φ preserves the triples ( I, J, M ) appearing in Propositions 4.8 and 4.29. Thenthe restriction of φ to antichains in the root poset (i.e. those with J = ∅ ) is abijection from antichains in the root poset to c -sortable elements. Remark 4.31.
The methods of this section don’t apply well to the case where c isnot bipartite, because the main structural results of the section, Propositions 4.18and 4.27, can fail when c is not bipartite. We now describe how both propositionsfail for linear c in type A . By analogy to Proposition 3.5, the c -bisortable elementsfor linear c are in bijection with noncrossing arc diagrams such that every arc eitherpasses only left of points or passes only right of points. (Each arc in the diagramcorresponds to a canonical joinand. See Remark 3.13 or [37, Example 4.10].) Tak-ing w = 3241 and u as in Proposition 4.18, the noncrossing arc diagram δ ( w ) hasa right arc connecting 1 to 4 and an arc (which is both a left arc and a right arc)connecting 2 to 3. Thus Can( w ) ∩ S = { s } . Also u = w = 3241, so Can( u ) \ S corresponds to the arc connecting 1 to 4, which has support { s , s , s } , contradict-ing Proposition 4.18(3). The c -sortable elements are in bijection with noncrossingarc diagrams such that every arc only passes right of points. From there, we easilysee that there is only 1 positive, clever c -sortable element, contradicting Proposi-tion 4.27(2).4.6. BiCatalan and Catalan formulas.
In this section and the next, we prepareto prove the formula for biCat( D n ) in Theorem 1.4, thus completing the proof ofthat theorem. Specifically, the proof requires combining a very large number ofidentities relating q -analogs of biCatalan numbers, Catalan numbers, and double-positive Catalan numbers that we quote or prove here. In this section, we giverecursions for the q -analogs of W -biCatalan and W -Catalan numbers for irreduciblefinite Coxeter groups, in which q -analogs of double-positive Catalan numbers appearas coefficients. Proposition 4.32.
For an irreducible finite Coxeter group W and a simple gen-erator s ∈ S , the q -analog of the W -biCatalan number satisfies (4.13) biCat( W ; q ) = (1 + q ) biCat( W S \{ s } ; q )+ 2 (cid:88) S Cat ++ ( W S ; q ) m (cid:89) i =1 (cid:20)
12 biCat( W S i ; q ) + 1 + q W S i \{ s i } ; q ) (cid:21) , where the sum is over all connected subgraphs S of the diagram for W with s ∈ S ,the connected components of the complement of S in the diagram are S , . . . , S m ,and each s i is the unique vertex in S i that is connected by an edge to a vertex in S .Proof. For fixed s , we break the formula in Theorem 4.2 into four sums, accordingto whether s is in S \ ( I ∪ J ∪ M ), in M , in I , or in J . The sum of terms with s ∈ S \ ( I ∪ J ∪ M ) equals biCat( W S \{ s } ; q ). The sum of terms with s ∈ M equals q · biCat( W S \{ s } ; q ).Consider next the sum of terms with s ∈ I , and in each term let S be theconnected component of the diagram containing s . Using (4.12), we can reorganizethe sum according to S to obtain (cid:88) S Cat ++ ( W S ; q ) (cid:88) q | M | Cat ++ ( W I (cid:48) ; q ) Cat ++ ( W J ; q ) , where the S -sum is as described in the statement of the proposition and the innersum is over all ordered triples ( I (cid:48) , J, M ) of disjoint subsets of S \ S such that noelement of I (cid:48) is connected by an edge of the diagram to an element of S . Againusing (4.12), we factor the inner sum further to obtain (cid:88) S Cat ++ ( W S ; q ) m (cid:89) i =1 (cid:104)(cid:88) q | M i | Cat ++ ( W I i ; q ) Cat ++ ( W J i ; q ) (cid:105) , where the S i and s i are as in the statement of the proposition and the inner sumruns of over all ordered triples ( I i , J i , M i ) of pairwise disjoint subsets of S i with s i (cid:54)∈ I i . The sum for each i can be broken up into a sum over terms with s i ∈ J i and terms with s i (cid:54)∈ J i . Splitting the sum over terms with s i (cid:54)∈ J i in half, we obtainthree sums: (cid:88) s i ∈ J i q | M i | Cat ++ ( W I i ; q ) Cat ++ ( W J i ; q )+ 12 (cid:88) s i (cid:54)∈ J i q | M i | Cat ++ ( W I i ; q ) Cat ++ ( W J i ; q )+ 12 (cid:88) s i (cid:54)∈ J i q | M i | Cat ++ ( W I i ; q ) Cat ++ ( W J i ; q )The symmetry between I and J on the right side of Theorem 4.2 lets us recognizethe sum of the first two terms as biCat( W S i ; q ), recalling that s (cid:54)∈ I i throughout.The third term is q biCat( W S i \{ s i } ; q ). We see that the sum of terms with s ∈ I is the sum in the proposed formula, without the factor 2 in front. By symmetry,the sum of terms with s ∈ J is the same sum, so we obtain the factor 2 in the sumand we have established the desired formula. (cid:3) We obtain the following recursion for biCat( D n ; q ) from Proposition 4.32. Thenotation D means A × A and D means A . Proposition 4.33.
For n ≥ , (4.14) biCat( D n ; q ) = (1 + q ) biCat( D n − ; q )+ n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:0) biCat( D n − i ; q ) + (1 + q ) biCat( D n − i − ; q ) (cid:1) + 2(1 + q ) Cat ++ ( A n − ; q ) + 4(1 + q ) Cat ++ ( A n − ; q ) + 2 Cat ++ ( D n ; q ) Proof.
In Proposition 4.32, take s to be a leaf of the D n diagram whose removalleaves the diagram for D n − . The sum over S splits into several pieces. First, the S for which the diagram on S \ { S } is of type D k for k ≥ (cid:80) n − i =1 Cat ++ ( A i ; q ) (cid:0) biCat( D n − i ; q ) + (1 + q ) biCat( D n − i − ; q ) (cid:1) . Next, the term forwhich the diagram on S \ { S } is of type D is 2 Cat ++ ( A n − )( (1 + q ) + q · ,which simplifies to 2(1 + q ) Cat ++ ( A n − ). The two terms for which the diagramon S \ { S } is of type A each contribute 2(1 + q ) Cat ++ ( A n − ). Finally, the termwith S = S is 2 Cat ++ ( D n ; q ). (cid:3) We obtain the following recursion for biCat( B n ; q ) from Proposition 4.32 simi-larly. Here and throughout the paper, we interpret B and B to be synonyms for A and A . OXETER-BICATALAN COMBINATORICS 45
Proposition 4.34.
For n ≥ , (4.15) biCat( B n ; q ) =(1 + q ) biCat( B n − ; q ) + 2 Cat ++ ( B n ; q ) + 2(1 + q ) Cat ++ ( A n − ; q )+ n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:2) biCat( B n − i ; q ) + (1 + q ) biCat( B n − i − ; q ) (cid:3) . Proof.
In Proposition 4.32, take s to be a leaf of the B n diagram whose removalleaves the diagram for B n − . The terms with | S | from 1 to n − | S | = n − | S | = n .For the term with | S | = n −
1, we use the facts that biCat( B ; q ) = (1 + q ) andthat biCat( B ; q ) = 1. (cid:3) Similarly, we obtain the following recursion for biCat( A n ) by taking s to beeither leaf of the diagram. Proposition 4.35.
For n ≥ , (4.16) biCat( A n ; q ) = (1 + q ) biCat( A n − ; q ) + 2 Cat ++ ( A n ; q )+ n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:2) biCat( A n − i ; q ) + (1 + q ) biCat( A n − i − ; q ) (cid:3) . Next we gather some formulas involving the q -Catalan numbers. We begin withthe usual recursion for the type-A Catalan numbers, although this q -version may beless widely familiar. It is easily obtained through the interpretation of Cat( A n ; q ) asthe descent generating function for 231-avoiding permutations in S n +1 , by breakingup the count according to the first entry in the permutation. We omit the details. Proposition 4.36.
For n ≥ , (4.17) Cat( A n ; q ) = (1 + q ) Cat( A n − ; q ) + q n − (cid:88) i =1 Cat( A i − ; q ) Cat( A n − i − ; q ) . Furthermore, using known formulas for the Narayana numbers, we obtain arecursion that relates the q -Catalan number in types A and D. Proposition 4.37.
For n ≥ , (4.18) Cat( D n ; q ) = n + 12 (1+ q ) Cat( A n − ; q ) − (cid:16) n −
12 + q + n − q (cid:17) Cat( A n − ) . Proof.
Taking the coefficient of q k on both sides, we see that (4.18) is equivalent to(4.19) Nar k ( D n ) = n + 12 (cid:0) Nar k ( A n − ) + Nar k − ( A n − ) (cid:1) − n −
12 Nar k ( A n − ) − Nar k − ( A n − ) − n −
12 Nar k − ( A n − ) . This can be verified using the known formulas for the type-A and type-D Narayananumbers. (See for example in [19, (9.1)] and [19, (9.3)], putting m = 1 in bothformulas). (cid:3) Next, we give a recursion for Cat( W ; q ) analogous to (4.13). The proof follows theoutline of the proof of Proposition 4.32, using Theorem 4.9 instead of Theorem 4.2.This proof is simpler than the proof of Proposition 4.32, so we omit the details. Proposition 4.38.
For an irreducible finite Coxeter group W and a simple gen-erator s , the q -analog of the W -Catalan number satisfies (4.20) Cat( W ; q ) = (1 + q ) Cat( W S \{ s } ; q )+ (cid:88) S Cat ++ ( W S ; q ) m (cid:89) i =1 (1 + q ) Cat( W S i \{ s i } ; q ) , where the sum is over all connected subgraphs S of the diagram for W with s ∈ S ,the connected components of the complement of S in the diagram are S , . . . , S m ,and each s i is the unique vertex in S i that is connected by an edge to a vertex in S . The following three propositions give the type-A, type-B, and type-D cases of(4.20).
Proposition 4.39.
For n ≥ , (4.21) Cat( A n ; q ) = Cat ++ ( A n ; q ) + (1 + q ) n − (cid:88) i =0 Cat ++ ( A i ; q ) Cat( A n − i − ; q ) . Proof. If n = 0, then the formula is Cat( A ; q ) = Cat ++ ( A ; q ), which says 1 = 1.Otherwise, taking s to be a leaf of the A n diagram in (4.20), the sum over S has terms (cid:80) n − i =1 Cat ++ ( A i ; q )(1 + q ) Cat( A n − i − ; q ) and Cat ++ ( A n ; q ). BecauseCat ++ ( A ; q ) = 1, we can merge the first term into the sum. (cid:3) Proposition 4.40.
For n ≥ , (4.22) Cat( B n ; q ) = Cat ++ ( B n ; q ) + (1 + q ) n − (cid:88) i =0 Cat ++ ( A i ; q ) Cat( B n − i − ; q ) Proof.
The formula holds for n = 0 and n = 1. For n >
1, take s to be the leafwhose deletion leaves a diagram of type B n − in (4.20), and rearrange the formulaas in the proof of Proposition 4.21. (cid:3) Proposition 4.41.
For n ≥ , (4.23) Cat( D n ; q ) =(1 + q ) Cat( A n − ; q ) + (1 + q ) Cat ++ ( A n − ; q ) + Cat ++ ( D n ; q )+ (1 + q ) n − (cid:88) i =1 Cat ++ ( A i ; q ) Cat( A n − i − ; q )+ (1 + q ) n − (cid:88) i =3 Cat ++ ( D i ; q ) Cat( A n − i − ; q ) . Proof.
Start with Proposition 4.38, taking s to be a leaf of the D n diagram whoseremoval leaves the diagram for A n − . For S not containing the leaf symmetric to s ,we get terms (1 + q ) (cid:80) n − i =1 Cat ++ ( A i ; q ) Cat( A n − i − ; q ) and (1 + q ) Cat ++ ( A n − ; q ).(The i = 1 term in the sum would be wrong, except that Cat ++ ( A ) = 0.) For S containing the leaf symmetric to s , we get (1+ q ) (cid:80) n − i =3 Cat ++ ( D i ; q ) Cat( A n − i − ; q )and Cat ++ ( D n ; q ). (cid:3) OXETER-BICATALAN COMBINATORICS 47
The double-positive Catalan numbers.
In this section, we consider thedouble-positive Catalan numbers for the classical reflection groups, and establishsome identities for Cat ++ ( A n ; q ), Cat ++ ( B n ; q ), and Cat ++ ( D n ; q ) that will be usefulfor proving the type-D case of Theorem 1.4. As an aside at the end of this section,we give formulas and computed values of the numbers Cat ++ ( W ) for all finite types.(See Theorem 4.52.) Remark 4.42.
Athanasiadis and Savvidou, in [3, Theorem 1.2], gave formulas forthe polynomials Cat ++ ( W ; q ) for each W of finite type by explicitly determiningcoefficients ξ i such that Cat ++ ( W ; q ) = (cid:80) (cid:98) n/ (cid:99) i =0 ξ i q i (1 + q ) n − i . Similar formulas forthe relevant polynomials Cat( W ; q ) are known [30, Propositions 11.14–11.15], sothe identities we need can in principle be obtained by manipulating the formulasfrom [3, 30]. Indeed, Proposition 4.44 is easily obtained in this way, but such proofsof Propositions 4.43 and 4.51 appear to be more complicated.To motivate the propositions that follow, we list here some examples of thedouble-positive Catalan numbers for the classical reflection groups. W A A A A A A A B B B B B D D D D Cat ++ ( W ) 1 0 1 2 6 18 57 2 6 22 80 296 10 42 168 660 From inspection of these numbers, several interesting relationships appear. First,the data suggests that 2 Cat ++ ( A n ) + Cat ++ ( A n − ) = Cat( A n − ). Below, we estab-lish a q -analog of this identity. Proposition 4.43.
For n ≥ , (4.24) (1 + q ) Cat ++ ( A n ; q ) + q Cat ++ ( A n − ; q ) = q Cat( A n − ; q ) . Proof. If n = 1, then the identity is (1 + q ) · q · q ·
1. If n >
1, then byinduction, we can replace (1+ q ) Cat ++ ( A i ; q ) with q (Cat( A i − ; q ) − Cat ++ ( A i − ; q ))in the terms i > ++ ( A ; q ) = 1 to obtainCat( A n ; q ) = Cat ++ ( A n ; q ) + (1 + q ) Cat( A n − ; q )+ q n − (cid:88) i =1 Cat( A i − ; q ) Cat( A n − i − ; q ) − q n − (cid:88) i =1 Cat ++ ( A i − ; q ) Cat( A n − i − ; q ) . The first sum, by Proposition 4.36, is (Cat( A n ; q ) − (1 + q ) Cat( A n − ; q )). Thesecond sum can be reindexed to q (cid:80) n − i =0 Cat ++ ( A i ; q ) Cat( A n − i − ; q ), which, byProposition 4.39, equals q q (Cat( A n − ; q ) − Cat ++ ( A n − ; q )). We obtainCat( A n ; q ) = Cat ++ ( A n ; q ) + (1 + q ) Cat( A n − ; q )+ Cat( A n ; q ) − (1 + q ) Cat( A n − ; q ) − q q (Cat( A n − ; q ) − Cat ++ ( A n − ; q )) , which simplifies to the desired identity. (cid:3) The data also suggests that Cat ++ ( D n ) = ( n −
2) Cat( A n − ). Indeed, the fol-lowing is a q -analog. Proposition 4.44.
For n ≥ , (4.25) Cat ++ ( D n ; q ) = ( n − q Cat( A n − ; q ) . Proof.
For n = 2, the identity is q + q = (3 − q (1 + q ). If n ≥
3, then we startwith (4.23). The first summation in the formula can be rewritten, using (4.21), as(4.26) (1 + q ) (cid:0) Cat( A n − ; q ) − (1 + q ) Cat( A n − ; q ) − Cat ++ ( A n − ; q ) (cid:1) . By induction, the second summation can be rewritten as(4.27) (1 + q ) q n − (cid:88) i =3 ( i −
2) Cat ++ ( A i − ; q ) Cat( A n − i − ; q ) . To further simplify (4.27), we use (4.17) to calculate( n − (cid:0) Cat( A n − ; q ) − (1 + q ) Cat( A n − ; q ) (cid:1) = ( n − q n − (cid:88) i =1 Cat( A i − ; q ) Cat( A n − i − ; q )= q n − (cid:88) i =1 (cid:0) ( i −
1) Cat( A i − ; q ) Cat( A n − i − ; q )+ ( n − i −
2) Cat( A i − ; q ) Cat( A n − i − ; q ) (cid:1) = q n − (cid:88) i =1 ( i −
1) Cat( A i − ; q ) Cat( A n − i − ; q )+ q n − (cid:88) i =1 ( n − i −
2) Cat( A i − ; q ) Cat( A n − i − ; q )Both sums can be reindexed to agree with (4.27), except for the initial factor (1+ q ).Thus (4.27) equals n − (1 + q )(Cat( A n − ; q ) − (1 + q ) Cat( A n − ; q )). Finally, we use(4.18) to rewrite the Cat( D n ; q ). We obtain(4.28) n + 12 (1 + q ) Cat( A n − ; q ) − (cid:16) n −
12 + q + n − q (cid:17) Cat( A n − ) =(1 + q ) Cat( A n − ; q ) + (1 + q ) Cat ++ ( A n − ; q ) + Cat ++ ( D n ; q )+ (1 + q ) (cid:0) Cat( A n − ; q ) − (1 + q ) Cat( A n − ; q ) − Cat ++ ( A n − ; q ) (cid:1) + n −
32 (1 + q )(Cat( A n − ; q ) − (1 + q ) Cat( A n − ; q )) . This can be rearranged to say Cat ++ ( D n ; q ) = ( n − q Cat( A n − ; q ). (cid:3) In order to establish a needed identity for double-positive Catalan numbers oftype B, we need a recursion for the q -Catalan number that comes from a completelydifferent direction. The q -Catalan numbers Cat( W ; q ) encode the h -vector of thegeneralized associahedron for W . (See, for example, [18, Section 5.2].) For eachCoxeter group W of rank n and each i from 0 to n , define f i to be the number ofsimplices in the simplicial generalized associahedron having exactly i vertices (andthus dimension i − f ( W ; x ) = n (cid:88) k =0 f k ( W ) x k . OXETER-BICATALAN COMBINATORICS 49
The following is [20, Proposition 3.7].
Proposition 4.45. If W is reducible as W × W , then f ( W ; x ) = f ( W ; x ) f ( W ; x ) .If W is irreducible with Coxeter number h , then (4.29) d f ( W ; x )d x = h + 22 (cid:88) s ∈ S f ( W S \{ s } ; x )Since f ( W ) encodes the f -vector of the generalized associahedron and Cat( W ; q )encodes the h -vector, (4.29) implies a formula for Cat( W ; q ). Since f ( W ) is hascoefficients reversed from the f -polynomial usually used to define h -vectors, theformula for Cat( W ; q ) is somewhat more complicated than (4.29). Proposition 4.46.
For an irreducible Coxeter group W with rank n ≥ andCoxeter number h , the q -analog of the Catalan number satisfies (4.30) n Cat( W ; q ) + (1 − q ) dd q Cat( W ; q ) = h + 22 (cid:88) s ∈ S Cat( W S \{ s } ; q ) . Proof.
We begin with the right side of (4.30) and replace q by x + 1 throughout.The result is h +22 (cid:80) s ∈ S rev( f ( W S \{ s } ; x )), where rev is the operator that reversesthe coefficients of a polynomial. In other symbols: x n − h +22 (cid:80) s ∈ S f ( W S \{ s } ; x − )Using (4.29), the quantity becomes x n − d f ( W ; x − )d( x − ) .Similarly, Cat( W ; x + 1) = rev( f ( W ; x )) = x n f ( W ; x − ), so f ( W ; x − ) = x − n Cat( W ; x + 1). Thus the right side of (4.30) equals x n − dd( x − ) (cid:2) x − n Cat( W ; x + 1) (cid:3) = x n − dd x (cid:2) x − n Cat( W ; x + 1) (cid:3) ( − x )= − x n +1 (cid:20) − nx − n − Cat( W ; x + 1) + x − n dd x Cat( W ; x + 1) (cid:21) = n Cat( W ; x + 1) − x dd x Cat( W ; x + 1)= n Cat( W ; x + 1) − x dd( x + 1) Cat( W ; x + 1)Replacing x by q − (cid:3) The type-B version of (4.30) is the following recursion:
Proposition 4.47.
For n ≥ , (4.31) n Cat( B n ; q )+(1 − q ) dd q Cat( B n ; q ) = ( n +1) n (cid:88) i =1 Cat( A i − ; q ) Cat( B n − i ; q ) . The following formula is obtained using known formulas for Narayana numbersof types A and B.
Proposition 4.48.
For n ≥ , (4.32) n (cid:88) i =1 Cat( A i − ; q ) Cat( B n − i ; q ) = n Cat( A n − ; q ) . Proof.
By (4.31), the assertion is equivalent to(4.33) n Cat( B n ; q ) + (1 − q ) dd q Cat( B n ; q ) = n ( n + 1) Cat( A n − ; q ) . Taking the coefficient of q k on both sides, we see that (4.33) is equivalent to(4.34) ( n − k ) Nar k ( B n ) + ( k + 1) Nar k +1 ( B n ) = n ( n + 1) Nar k ( A n − ) . This can be verified using the formulas for the type-A and type-B Narayana num-bers, found for example in [19, (9.1)] and [19, (9.2)] (setting m = 1 in both formulasfrom [19]). (cid:3) Using (4.16) and the observation that biCat( A n ; q ) = Cat( B n ; q ), then applying(4.22) twice, (where, in the first instance n is replaced by n + 1 in (4.22)), we obtainthe following formula. Proposition 4.49.
For n ≥ , (4.35) Cat( B n ; q ) = (1 + q ) Cat( B n − ; q ) − (1 + q ) Cat ++ ( A n − ; q )+ Cat ++ ( B n ; q ) + (1 + q ) Cat ++ ( B n − ; q ) . Next, we obtain the following formula.
Proposition 4.50.
For n ≥ , (4.36) (1 + q ) Cat( B n ; q ) =(1 + q + q ) Cat( B n − ; q ) + ( n − q (1 + q ) Cat( A n − ; q )+ q Cat ++ ( B n − ; q ) + (1 + q ) Cat ++ ( B n ; q ) . Proof.
Using (4.24) to replace each instance of (1 + q ) Cat ++ ( A i ; q ) in (4.22) with q (Cat( A i − ; q ) − Cat ++ ( A i − ; q )) for i > B n ; q ) = (1 + q ) Cat( B n − ; q ) + q n − (cid:88) i =1 Cat( A i − ; q ) Cat( B n − i − ; q ) − q n − (cid:88) i =1 Cat ++ ( A i − ; q ) Cat( B n − i − ; q ) + Cat ++ ( B n ; q )We use (4.32) with n replaced by n − n replaced by n − B n ; q ) = (1 + q ) Cat( B n − ; q ) + q ( n −
1) Cat( A n − ; q ) − q q (Cat( B n − ; q ) − Cat ++ ( B n − ; q )) + Cat ++ ( B n ; q ) . We multiply through by (1 + q ) and simplify to obtain (4.36). (cid:3) Solving both (4.36) and (4.35) for (1+ q ) Cat ++ ( B n ; q ) and combining them, thensolving for (1 + q + q ) Cat ++ ( B n − ; q ), we obtain the key result for Cat ++ ( B n − ). Proposition 4.51. (4.37) (1 + q + q ) Cat ++ ( B n − ; q ) = − q Cat( B n − ; q )+ ( n − q (1 + q ) Cat( A n − ; q ) + (1 + q ) Cat ++ ( A n − ; q ) OXETER-BICATALAN COMBINATORICS 51
We have established all the propositions we will need for the type-D case of The-orem 1.4. We now provide formulas and values for the double-positive W -Catalannumbers, although they are not necessary for the Coxeter-biCatalan combinatorialresults of this paper. Theorem 4.52.
The double-positive W -Catalan numbers are W A n ( n ≥ B n ( n ≥ ++ ( W ) (cid:80) nk =0 ( − k k +1 n +1 (cid:0) n − kn (cid:1) (cid:80) nk =0 ( − k (cid:0) n − k − n − (cid:1) D n ( n ≥ E E E F H H I ( m ) n − n (cid:0) n − n − (cid:1)
265 1728 13816 49 16 208 m − ++ ( A n ) are OEIS [28] sequence number A000957(the Fine numbers). The type-B and type-D double-positive Catalan numbers aresequences A014301 and A276666. Proof.
The proof for Cat ++ ( A n ) can be found for example in [14, Sections 3–4],where both the q = 1 specialization of (4.24) and the formula (cid:80) nk =0 ( − k k +1 n +1 (cid:0) n − kn (cid:1) are shown to describe the Fine numbers.To obtain the formula for Cat ++ ( B n ), we start from (4.37), set q = 1 and shiftthe index n to obtain(4.38) 3 Cat ++ ( B n ) = − Cat( B n ) + 2 n Cat( A n − ) + 4 Cat ++ ( A n )We use the q = 1 specialization of (4.24) to rewrite 4 Cat ++ ( A n ), as 2 Cat( A n − ) − ++ ( A n − ), use known formulas for Cat( B n ), Cat( A n − ), and Cat ++ ( A n − ),and reindex a sum to obtain(4.39) 3 Cat ++ ( B n ) = (cid:18) nn (cid:19) + 2 n (cid:88) k =0 ( − k kn (cid:18) n − k − n − (cid:19) Verifying the desired formula for Cat ++ ( B n ) can thus be reduced to verifying theidentity(4.40) n (cid:88) k =0 ( − k n − kn (cid:18) n − k − n − (cid:19) = (cid:18) nn (cid:19) . Rewriting n − kn (cid:0) n − k − n − (cid:1) as 2 (cid:0) n − kn (cid:1) − (cid:0) n − k − n − (cid:1) = (cid:0) n − kn (cid:1) + (cid:0) n − k − n (cid:1) , we see thatthe alternating sum telescopes, collapsing to (cid:0) nn (cid:1) as desired.The formula for Cat ++ ( D n ) follows from (4.25) and the usual formula for Cat( A n ).The values for the exceptional types are easily computed. (cid:3) The Type D biCatalan number.
We now complete the proof of Theo-rem 1.4 by proving the following theorem.
Theorem 4.53.
For n ≥ , the D n -biCatalan number is (4.41) biCat( D n ) = 6 · n − − (cid:18) n − n − (cid:19) . Since we have already established the type-A and type-B cases of Theorem 1.4,Theorem 4.53 is the assertion that biCat( D n ) = 3 biCat( B n − ) − A n − ). In preparation for the proof, we let X = X ( q ) and Y = Y ( q ) be any rational functionsof q and define, for each n ≥
2, a rational function Z n = Z n ( q ) given by Z n = biCat( D n ; q ) − X biCat( B n − ; q ) + Y biCat( A n − ; q ) . Combining (4.14), (4.15), and (4.16), we obtain the following recursion for Z n for n ≥ Z n = (1 + q ) Z n − + n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:0) Z n − i + (1 + q ) Z n − i − (cid:1) + 2 (cid:0) (1 + q ) − X (1 + q ) + Y (cid:1) Cat ++ ( A n − ) + 4(1 + q ) Cat ++ ( A n − )+ 2 Cat ++ ( D n ; q ) − X Cat ++ ( B n − ; q )One way to obtain a formula for q -biCatalan numbers biCat( D n ; q ) would be tofind a choice of X and Y that makes this recursion for Z n into something that canbe solved. We have thus far been unable to find a choice of X and Y that works.Instead, we will prove Theorem 4.53 by showing that if X (1) = 3 and Y (1) = 2,then Z n (1) = 0 for all n ≥
2. In the proof that follows, we take convenient choicesof X and Y but delay specializing q to 1 until the end, because specializing earlierdoes not make the manipulations much easier, and because we hope that perhapswe are still getting closer to a formula for biCat( D n ; q ). Proof of Theorem 4.53.
Substituting (4.37) and (4.25) into (4.42), taking X to be1 + q + q , and taking Y to be 2 q − q + q , we obtain(4.43) Z n = (1 + q ) Z n − + n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:0) Z n − i + (1 + q ) Z n − i − (cid:1) + 2 q (1 − q ) Cat ++ ( A n − ; q ) + 2(1 − q )(1 + q ) Cat ++ ( A n − ; q ) − q (cid:0) n − q (cid:1) Cat( A n − ; q ) + 2 q Cat( B n − ; q )We next apply (4.24) to rewrite the two double-positive q -Catalan numbers in (4.43)as a single q -Catalan number.(4.44) Z n = (1 + q ) Z n − + n − (cid:88) i =1 Cat ++ ( A i ; q ) (cid:0) Z n − i + (1 + q ) Z n − i − (cid:1) + 2 q (1 − q ) Cat( A n − ; q ) − q (cid:0) n − q (cid:1) Cat( A n − ; q ) + 2 q Cat( B n − ; q )Finally specializing q to 1 and using the fact that Cat( B n − ) = n Cat( A n − ) for n ≥ Z n (1) = 2 Z n − (1) + n − (cid:88) i =1 Cat ++ ( A i ) (cid:0) Z n − i (1) + 2 Z n − i − (1) (cid:1) We easily verify that Z (1) = 0, and thus we have a simple inductive proof that Z n (1) = 0 for all n ≥
2. Since we chose X and Y to have X (1) = 3 and Y (1) = 2,we obtain the desired identity biCat( D n ) = 3 biCat( B n − ) − A n − ). (cid:3) OXETER-BICATALAN COMBINATORICS 53
Type-D biNarayana numbers.
Computational evidence suggests the fol-lowing modest conjecture on the type-D biNarayana number biNar k ( D n ). Conjecture 4.54.
The type-D biNarayana number biNar k ( D n ) is a polynomialin n (for n ≥ ) of degree k and leading coefficient k (2 k )! . If Conjecture 4.54 is true, then the following table shows (2 k )!2 k · biNar k ( D n ) forsmall k . k (2 k )!2 k · biNar k ( D n )0 11 2 n − n n − n + 35 n − n −
243 8 n − n + 365 n − n + 212 n + 1104 n − n − n + 2268 n − n + 20349 n − n − n + 104826 n − k = 1 case is verified by Proposition 2.4, and with some effort, the k = 2 casecan be proved as well. Acknowledgments
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