A unifying framework for the ν-Tamari lattice and principal order ideals in Young's lattice
Matias von Bell, Rafael S. González D'León, Francisco A. Mayorga Cetina, Martha Yip
aa r X i v : . [ m a t h . C O ] F e b A UNIFYING FRAMEWORK FOR THE ν -TAMARI LATTICE ANDPRINCIPAL ORDER IDEALS IN YOUNG’S LATTICE MATIAS VON BELL, RAFAEL S. GONZ ´ALEZ D’LE ´ON, FRANCISCO A. MAYORGA CETINA,AND MARTHA YIP
Abstract.
We present a unifying framework in which both the ν -Tamari lattice, intro-duced by Pr´eville-Ratelle and Viennot, and principal order ideals in Young’s lattice indexedby lattice paths ν , are realized as the dual graphs of two combinatorially striking triangu-lations of a family of flow polytopes which we call the ν -caracol flow polytopes. The firsttriangulation gives a new geometric realization of the ν -Tamari complex introduced by Ce-ballos, Padrol and Sarmiento. We use the second triangulation to show that the h ∗ -vectorof the ν -caracol flow polytope is given by the ν -Narayana numbers, extending a result ofM´esz´aros when ν is a staircase lattice path. Our work generalizes and unifies results on thedual structure of two subdivisions of a polytope studied by Pitman and Stanley. Key words and phrases : flow polytope, triangulation, ν -Dyck path, ν -Tamari lattice,Young’s lattice Introduction
Flow polytopes are a family of beautiful mathematical objects. They appear in optimiza-tion theory as the feasible sets in maximum flow problems and they also appear in otherareas of mathematics including representation theory and algebraic combinatorics. In thefollowing, G = ( V, E ) denotes a connected directed graph with vertex set V = { , , . . . , n +1 } and edge multiset E with m edges, with n, m ∈ N . We assume that any edge ( i, j ) ∈ E isdirected from i to j whenever i < j and hence G is acyclic . At each vertex i ∈ V we assigna net flow a i ∈ Z satisfying the balance condition P n +1 i =1 a i = 0, and hence a n +1 = − P ni =1 a i .For a = ( a , . . . , a n , − P ni =1 a i ) ∈ Z n +1 , an a -flow on G is a tuple ( x e ) e ∈ E ∈ R m ≥ such that X e ∈ out( j ) x e − X e ∈ in( j ) x e = a j where in( j ) and out( j ) respectively denote the set of incoming and outgoing edges at j , for j = 1 , . . . , n . In what follows, by a graph G we mean a connected directed acyclic graphwhose sets out(1), in( n + 1), and in( j ) and out( j ) for j = 2 , . . . , n , are nonempty. The flowpolytope of G with net flow a is the set F G ( a ) of a -flows on G . In this article we only considerflow polytopes with unitary flow a = e − e n +1 , where e i for i = 1 , . . . , n + 1 denotes thestandard basis in R n +1 , and we will abbreviate the flow polytope of G with unitary flow as F G . In this case, the only integral points of F G are its vertices, which correspond to theunitary flows along maximal directed paths of G from vertex 1 to n + 1. Such maximal pathsare called routes (see Figure 6).A d -simplex is the convex hull of d + 1 points in general position in R k with k ≥ d . A(lattice) triangulation of a d -polytope P is a collection T of d -simplices each of whose vertices are in P ∩ Z d , such that the union of the simplices in T is P , and any pair of simplices intersectin a (possibly empty) common face. The normalized volume of a d -polytope is d ! times itsEuclidean volume. Since the Euclidean volume of a unimodular d -simplex in R d is d ! , then byenumerating the simplices in a unimodular triangulation, one can compute the normalizedvolume of the polytope.Baldoni and Vergne [3] gave a set of formulas to determine the normalized volume of F G ( a ), and these are known as the Lidskii formulas. M´esz´aros and Morales [13] describeda triangulation approach due to Postnikov and Stanley, providing an alternative proof ofthe Lidskii formulas. Together with Striker [14], showed that a family of Postnikov–Stanleytriangulations, which they call framed, are equivalent to the triangulations introduced byDanilov, Karzanov and Koshevoy in [10], which depend on the notion of a framing on a graph(see Section 3). Different framings of a graph give different regular unimodular triangulationsof the associated flow polytope.The combinatorial structure of a triangulation T of a polytope is encoded in its dual graph .This is a graph on the set of simplices in T with edges between simplices sharing a commonfacet. In this article we introduce the family of ν -caracol graphs car( ν ) (see Definition 2.1).These graphs are indexed by lattice paths ν in Z , and are similar to constructions in [13]and [18]. In Sections 4 and 5 we discuss two particular framings on car( ν ) which we callthe length and the planar framings. The triangulations arising from these framings haveconnections to two lattices on ν -Catalan objects (see Section 2.1) that appear recurrently inthe literature:(1) The ν -Tamari lattice Tam( ν ) introduced by Pr´eville-Ratelle and Viennot [15].(2) The principal order ideals I ( ν ) determined by ν in Young’s lattice Y .Thus we find that the collection of framed triangulations on F car( ν ) provides a unifyingframework for studying these two ubiquitous lattice structures. The family of ν -Catalanobjects is a generalization of the classical Catalan and rational Catalan families of objectsthat have been extensively studied in the recent literature (see for example [1, 2, 7, 15]).Our main results are the following: Theorem 1.1.
The normalized volume of the flow polytope F car( ν ) is given by the number of ν -Dyck paths, that is, the ν -Catalan number Cat( ν ) . Theorem 1.2.
The length-framed triangulation of F car( ν ) is a regular unimodular triangu-lation whose dual graph is the Hasse diagram of the ν -Tamari lattice Tam( ν ) . Theorem 1.3.
The planar-framed triangulation of F car( ν ) is a regular unimodular triangu-lation whose dual graph is the Hasse diagram of the principal order ideal I ( ν ) in Young’slattice Y . Theorem 1.4.
The h ∗ -polynomial of F car( ν ) is the ν -Narayana polynomial. To describe the combinatorial structure of the length-framed and planar-framed triangula-tions we use three different ν -Catalan families of objects, as each highlights the combinatoricsin crucial and distinct ways. These are the ( I, J )-trees introduced by Ceballos, Padrol andSarmiento [8], the ν -Dyck paths introduced by Pr´eville-Ratelle and Viennot [15], and the UNIFYING FRAMEWORK 3 ν -trees which were also introduced by Ceballos et al. [9] (see Sections 4.1.2, 4.1.1, and 4.1.3respectively). The role that they play in the combinatorics of the triangulations is summa-rized in Table 1 below. The interested reader can visit [8] and [9] for the correspondencesbetween ( I, J )-trees, ν -trees and ν -Dyck paths and [5] for their generalizations to ( I, J )-forests, ν -Schr¨oder trees and ν -Schr¨oder paths. Triangulation Vertices Simplices Adjacency Dualgraph
Length-framed Arcs of (
I, J )-trees (
I, J )-trees Two (
I, J )-trees that mmdiffer by one arc Hasse diag.of Tam( ν )Lattice pointsabove ν ν -trees Two ν -trees that differ bya rotation(not obtaineddirectly) ν -Dyckpaths Two ν -Dyck paths thatdiffer by a rotationPlanar-framed Lattice pointsabove ν ν -Dyckpaths Two ν -Dyck paths thatdiffer by a pair EN to NE Hasse diag.of I ( ν ) Table 1. ν -Catalan objects and their role in the combinatorial structure ofthe two framed triangulations.We point out that the study of length-framed and planar-framed triangulations of flowpolytopes on the ν -caracol graphs can be extended systematically to all graphs. On the ν -caracol graphs, the length framing can be viewed as ordering both the incoming andoutgoing sets of edges at each inner vertex according to decreasing length, while the planarframing can be viewed as ordering the set of incoming edges at each inner vertex with respectto decreasing length while the set of outgoing edges is ordered with respect to increasinglength. Particularly for graphs which are symmetric with respect to the vertical axis, thenour viewpoint suggests that these two framings are in a sense dual to each other, so perhapsit is not surprising that both framings lead to combinatorially interesting triangulations ofthe flow polytope. In a forthcoming article we study these triangulations in the more generalsetting.In the classical case when ν = (1 , , . . . ,
1) =: (1 n ), the simultaneous appearance of theTamari lattice and I (1 n ) (also known as the lattice of filters of the type A root poset) inthe study of polytopes has been observed before in the work of Pitman and Stanley [17] onsubdivisions of a family of polytopes Π n ( ν ) (see Remark 2.3) for the case when ν = (1 n ).In particular, the canonical triangulation of the order polytope O ( Q n ), where Q n is theproduct of a 2-chain and an n -chain, was used to show that Π n (1 n ) has a subdivision whosedual structure is given by the lattice I (1 n ). A second subdivision of Π n (1 n ) was constructedin [17] whose dual structure is the Tamari lattice. Via the Cayley trick, M´esz´aros andMorales [13, Section 7] observed that there is an embedding of Π n ( ν ) in a polytope that isintegrally equivalent to F car( ν ) . Following their conclusions, our results imply that the twotriangulations of F car( ν ) in Theorems 1.2 and 1.3 induce two subdivisions of Π n ( ν ) whosedual graphs are the Hasse diagrams of the ν -Tamari lattice and I ( ν ) respectively, therebygeneralizing the result obtained in [17] for the classical case ν = (1 n ). Our results alsounify the techniques used to exhibit these triangulations since both are obtained directly asframed-triangulations of F car( ν ) . VON BELL, GONZ ´ALEZ D’LE ´ON, MAYORGA CETINA, AND YIP
The results of M´esz´aros, Morales and Striker in [14] imply that the polytope F car( ν ) isintegrally equivalent to an order polytope O ( Q ν ) and that the planar-framed triangulationof F car( ν ) corresponds to the canonical triangulation of O ( Q ν ) under this equivalence. Thisalso generalizes the relation between Π n ( ν ) and O ( Q ν ) (see Figure 11) in the classical casewhen ν = (1 n ).This article is organized as follows. In Section 2 we introduce the ν -caracol graph car( ν )and its associated flow polytope F car( ν ) , proving in Theorem 1.1 that its normalized volumeis given by the ν -Catalan number Cat( ν ). In Section 3 we describe the theory of framedtriangulations as presented in [14]. In Section 4 we define the length framing of car( ν ),prove Theorem 1.2, and as consequence we conclude that the associated triangulation is ageometric realization of the ν -Tamari complex. In Section 5 we define the planar framing ofcar( ν ) and prove Theorem 1.3. We explain the relationship to order polytopes, showing thatour unifying framework generalizes results of Pitman and Stanley in [17]. As an application,in Section 6 we use the dual graph of the planar-framed triangulation of F car( ν ) to obtainthe h ∗ -polynomial, which proves Theorem 1.4. This result also gives a new proof that the h -vector of the ν -Tamari complex is given by the ν -Narayana numbers.2. The family of ν -caracol flow polytopes In [6], the second and fourth authors studied the flow polytope of the caracol graph, whosenormalized volume is the number of Dyck paths from (0 ,
0) to ( n, n ), a Catalan number. Wenow extend this construction.2.1. ν -Dyck paths and ν -Catalan numbers. Let a, b be nonnegative integers, and let ν be a lattice path from (0 ,
0) to ( b, a ), consisting of a sequence of a north steps N = (0 , b east steps E = (1 , ν -Dyck path is a lattice path from (0 ,
0) to ( b, a ) that liesweakly above ν .When a and b are coprime positive integers and ν is the lattice path that borders thesquares which intersect the line y = ab x , this is the special case of the rational ( a, b ) -Dyckpath studied by Armstrong, Loehr and Warrington in [1] who showed that the number ofrational ( a, b )-Dyck paths is the ( a, b ) -Catalan number Cat( a, b ) = a + b (cid:0) a + ba (cid:1) . See Figure 4for an example of a (3 , a, b ) = ( n, n + 1), this is the case ofthe classical Catalan number Cat( n ) = n +1 (cid:0) n +1 n (cid:1) = n +1 (cid:0) nn (cid:1) . For general ν , the numberCat( ν ) of ν -Dyck paths is calculated by a determinantal formula which can be derived byan application of the Gessel–Viennot Lemma [11]:Cat( ν ) = det (cid:18) P a − jk =1 ν k j − i (cid:19)! ≤ i,j ≤ a − , but no closed-form positive formula is known. For more on ν -Dyck paths, see for exampleCeballos and Gonz´alez D’Le´on [7], or Pr´eville-Ratelle and Viennot [15].2.2. The ν -caracol graph.Definition 2.1. Let a, b be nonnegative integers, and let ν be a lattice path from (0 ,
0) to( b, a ) where ν = N E ν N E ν · · · N E ν a . The ν -caracol graph car( ν ) is the graph on the vertex UNIFYING FRAMEWORK 5 set [ a + 3], together with ν i copies of the edge (1 , i + 2) for i = 1 , . . . , a , the edges ( i, a + 3)for i = 2 , . . . , a + 1, and the edges ( i, i + 1) for i = 1 , . . . , a + 2.Note that in this construction, the graph car( ν ) has n + 1 := a + 3 vertices, and the in-degree in i of the vertex i in car( ν ) is in = 1, in i = ν i − + 1 for i = 3 , . . . , n and in n +1 = n − m of car( ν ) is computed by summing the in-degrees of its vertices, sothat m = n +1 X i =2 in i = 1 + a X i =1 ( ν i + 1) + ( a + 1) = 2 a + b + 2 . The (intrinsic) dimension of a flow polytope is given by dim F G = | E ( G ) | − | V ( G ) | + 1, sowe can conclude from this that dim F car( ν ) = m − n = a + b . (0 ,
0) (7 , ν = 1 ν = 3 ν = 0 ν = 1 ν = 2 1 2 3 4 5 6 7 8 Figure 1.
A lattice path ν = N E N E N E N E N E and its associated ν -caracol graph car( ν ).The flow polytope on the graph car( ν ) in the special case when ν = (1 n ) has previouslybeen studied by M´esz´aros [12] and by Benedetti et al. [6]. Remark 2.2.
The careful reader will notice that in Definition 2.1 of the graph car( ν ) wechose to use a lattice path ν that begins with an N step. This choice was made for convenienceof the presentation, and is not a true restriction. From the results in Section 3 one can verifythat the combinatorial structure of a framed triangulation of the flow polytope F car( ν ) isnot affected by changing the number of N steps at the beginning of ν (or by changing thenumber of E steps at the end of ν ). Hence without loss of generality and unless otherwisespecified, all lattice paths ν begin with at least one N step.Two integral polytopes P ⊆ R m and Q ⊆ R n are integrally equivalent if there exists anaffine transformation ϕ : R m → R n whose restriction to P preserves the lattice. That is, ϕ is a bijection between Z m ∩ aff( P ) and Z n ∩ aff( Q ). Integrally equivalent polytopes have thesame Ehrhart polynomial, and hence the same volume. Remark 2.3.
M´esz´aros and Morales [13, Corollary 6.17] have previously considered aclosely-related variant of the flow polytope F car( ν ) , denoted as Π ⋆a ( ν ) in their work. Theunderlying graph of the flow polytope Π ⋆a ( ν ) can be obtained from car( ν ) by deleting theedge (2 , n + 1) and contracting the edge (1 , F car( ν ) and Π ⋆a ( ν ) are integrally equivalent.They observed that the normalized volume of Π ∗ a ( ν ) is the number of lattice points inthe Pitman–Stanley polytope Π a ( ν ) = { y ∈ R a | P ki =1 y i ≤ P ki =1 ν i } , which is equal tothe number of ν -Dyck paths. In the next section, we obtain a direct proof of this result bygiving a combinatorial interpretation to the vector partitions enumerated by the Kostantpartition function in the generalized Lidskii formula. This method was first considered in [6]and further developed in [18]. VON BELL, GONZ ´ALEZ D’LE ´ON, MAYORGA CETINA, AND YIP
The volume of the ν -caracol flow polytope. We begin by defining the Kostantpartition function of a graph, and the special case of the Lidskii volume formula which wewill use.For i = 1 , . . . , n , we call α i = e i − e i +1 the i -th simple root . For each edge e = ( i, j ) of agraph G , let α e = e i − e j = α i + · · · + α j − , and Φ + G = { α e | e ∈ E ( G ) } will be called themultiset of positive roots associated to G .A vector partition of the vector v with respect to Φ + G is a decomposition of v into a non-negative linear combination of the positive roots associated to G . The Kostant partitionfunction of G evaluated at v , denoted by K G ( v ), is the number of vector partitions of v withrespect to Φ + G . Integral v -flows on G are equivalent to vector partitions of v , so the numberof integral v -flows on G , and hence the number of lattice points in F G ( v ), is K G ( v ).Let G be a graph on the vertex set { , . . . , n + 1 } . For i = 2 , . . . , n + 1, let u i = in i − i . Proposition 2.4 (Baldoni and Vergne [3, Theorem 38]) . Let G be a graph with n + 1 vertices and m edges. The normalized volume of the flow polytope F G with unitary net flow a = e − e n +1 is given by vol F G = K G ( v in ) , where v in = (0 , u , . . . , u n , − ( m − n − u n +1 )) . For flow polytopes of ν -caracol graphs with unitary netflow, the Kostant partition func-tion K car( ν ) ( v in ) has a simple combinatorial interpretation which we now describe. Thisgeneralizes the construction for the case ν = N E k − N E k · · · N E k considered in [18, Section2.4]. Definition 2.5.
Let ν = N E ν · · · N E ν a be a lattice path from (0 ,
0) to ( b, a ). An in-degreegravity diagram for the flow polytope F car( ν ) consists of a collection of dots and line segmentswith the following properties:(i) The dots are arranged in columns indexed by the simple roots α , . . . , α a +2 , with ν + · · · + ν j − dots in the column indexed by α j , and all dots are drawn justifiedupwards.(ii) Horizontal line segments may be drawn between dots in consecutive columns so thateach dot is incident to at most one line segment. A trivial line segment is a singletondot. All non-trivial line segments must contain a dot in the column indexed by α a +2 (that is, all line segments are justified to the right). Longer line segments appearabove shorter line segments.We denote the set of all in-degree gravity diagrams by G car( ν ) ( v in ). See Figure 2 for anexample of an in-degree gravity diagram.The proof of the following Lemma is analogous to the one in [6, Theorem 3.1] for out-degreegravity diagrams. See also [18]. Lemma 2.6.
There is a bijection between the set of vector partitions of v in with respect to Φ +car( ν ) and the set of in-degree gravity diagrams for the flow polytope F car( ν ) . Consequently, K car( ν ) ( v in ) = |G car( ν ) ( v in ) | . UNIFYING FRAMEWORK 7
Example 2.7.
Let ν = N E N EN N E N E . A vector partition of v in = 2 α + 3 α + 3 α +6 α + 7 α with respect to the positive roots in Φ +car( ν ) is v in = α (3 , + α (5 , + 2 α (6 , + α + 2 α + α + 2 α + 3 α . This vector partition is represented by the gravity diagram on the left of Figure 2.We are ready to make a connection from in-degree gravity diagrams to ν -Dyck paths. Lemma 2.8.
There is a bijection between the set G car( ν ) ( v in ) of in-degree gravity diagramsfor the flow polytope F car( ν ) and the set D ν of ν -Dyck paths.Proof. First recall that a ν -Dyck path is a lattice path in the rectangular grid from (0 ,
0) to( b, a ) that lies weakly above the path ν . Also recall that in an in-degree gravity diagram forcar( ν ), the column indexed by α k has ν + · · · + ν k dots, for k = 3 , . . . , a + 2. This is preciselythe number of squares in the row between the lines x = 0, y = k − y = k , and above ν .Therefore, given an in-degree gravity diagram Γ ∈ G car( ν ) ( v in ), we may rotate it 90 degreescounterclockwise and embed the array of dots into the squares of Z so that the dots in thecolumn indexed by α a +2 lie in the row of squares just above the line y = a , and the dots inthe first row of Γ lie in the column of squares just right of the line x = 0. By the previousobservation, we see that the dots of Γ occupy every square in Z between the lines x = 0, x = b and y = a + 1, and which lie above the path ν . See Figure 2 for an illustration.Line segments of the rotated embedded gravity diagram Γ are now vertical, and theyextend down from just above the top row of the rectangular grid. The lengths of thesevertical line segments are weakly decreasing from left to right, so the line segments of Γ definea unique ν -Dyck path that separates the dots in Γ which are incident to a line segment in Γ,from the dots which are not incident to any (proper) line segment in Γ. This constructiondefines a map Ξ : G car( ν ) ( v in ) → D ν .Conversely, any ν -Dyck path defines an in-degree gravity diagram Γ for F car( ν ) , whereevery dot of Γ that occupies a square that is above the ν -Dyck path is incident to a linesegment of Γ, and every dot of Γ that occupies a square that is below the ν -Dyck path isnot incident to any (proper) line segment of Γ. Therefore, Ξ is a bijection. (cid:3) α α α α α (0 ,
0) (7 , ν = 1 ν = 3 ν = 0 ν = 1 ν = 2 α α α α α Figure 2.
A gravity diagram (left) representing a vector partition of v in asso-ciated to car( ν ) for ν = N E N EN N E N E . The bijection Ξ of Theorem 1.1sends the gravity diagram to the ν -Dyck path via a 90 degree rotation (right). Theorem 1.1.
The normalized volume of the flow polytope F car( ν ) is given by the number of ν -Dyck paths, that is, the ν -Catalan number Cat( ν ) . VON BELL, GONZ ´ALEZ D’LE ´ON, MAYORGA CETINA, AND YIP
Proof.
Combining Proposition 2.4 and Lemmas 2.6 and 2.8, the normalized volume of F car( ν ) is vol F car( ν ) = K car( ν ) ( v in ) = |G car( ν ) ( v in ) | = |D ν | = Cat( ν ) . (cid:3) In Sections 4 and 5, we construct two regular unimodular triangulations for the flowpolytope F car( ν ) with combinatorially interesting dual graph structures, giving two moreproofs that the normalized volume of F car( ν ) is the number of ν -Dyck paths.3. Framed triangulations of a flow polytope
We now describe the family of triangulations defined by Danilov, Karzanov, and Ko-shevoy [10], interpreted as special cases of the Postnikov–Stanley triangulations describedby M´esz´aros, Morales and Striker in [14].We call inner vertices the vertices { , . . . , n } of a graph G on n + 1 vertices. A framing at the inner vertex i is a pair of linear orders ( ≺ in( i ) , ≺ out( i ) ) on the incoming and outgoingedges at i . A framed graph , denoted ( G, ≺ ), is a graph with a framing at every inner vertex.In Sections 4 and 5, we will consider two specific framings of the caracol graphs car( ν ),which lead to combinatorially interesting triangulations of F car( ν ) . An example of these twoframings is given in Figure 3. Figure 3.
Length (left) and planar (right) framings at the vertex 6 of G = car( ν ).For an inner vertex i of a graph G , let In( i ) and Out( i ) respectively denote the set ofmaximal paths ending at i and the set of maximal paths beginning at i . For a route R containing an inner vertex i , let Ri (respectively iR ) denote the maximal subpath of R ending (respectively beginning) at i . Define linear orders ≺ In( i ) and ≺ Out( i ) on In( i ) andOut( i ) as follows. Given paths R, Q ∈ In( i ), let j ≤ i be the smallest vertex after which Ri and Qi coincide. Let e R be the edge of R entering j and let e Q be the edge of Q entering j . Then R ≺ In( i ) Q if and only if e R ≺ in( j ) e Q . Similarly for R, Q ∈ Out( i ), let j ≥ i be thelargest vertex before which iR and iQ coincide. Then R ≺ Out( i ) Q if and only if e R ≺ out( j ) e Q .Two routes R and Q containing an inner vertex i are coherent at i if Ri and Qi are orderedthe same as iR and iQ . Routes R and Q are coherent if they are coherent at each commoninner vertex. A set of mutually coherent routes is a clique . For a maximal clique C , let ∆ C denote the convex hull of the vertices of F G corresponding to the unitary flows along theroutes in C . Proposition 3.1 (Danilov et al. [10]) . Let ( G, ≺ ) be a framed graph. Then { ∆ C | C is a maximal clique of ( G, ≺ ) } UNIFYING FRAMEWORK 9 is the set of the top dimensional simplices in a regular unimodular triangulation of F G . The length-framed triangulation
The goal of this section is to show that the flow polytope F car( ν ) has a regular unimodulartriangulation whose dual graph structure is given by the Hasse diagram of the ν -Tamarilattice. The triangulation in question arises as a DKK triangulation with the length-framing .We show that this length-framed triangulation is combinatorially equivalent to the ν -Tamaricomplex introduced by Ceballos, Padrol, and Sarmiento in [8].4.1. The ν -Tamari lattice. The ν -Tamari lattice Tam( ν ) was introduced by Pr´eville-Ratelle and Viennot [15, Theorem 1] as a partial order on the set of ν -Dyck paths. Usingan alternative description with ( I, J )-trees, Ceballos, Padrol and Sarmiento [8] realized the ν -Tamari lattice as the one-skeleton of a polyhedral complex known as the ν -associahedron K ν , which generalizes the classical associahedron. In [9], they gave further descriptions ofthe ν -Tamari lattice using ν -trees and ν -bracket vectors, proving a special case of Rubey’slattice conjecture. In [5] the first and fourth authors generalize ν -Dyck paths and ν -trees to ν -Schr¨oder paths and ν -Schr¨oder trees in their study of the face poset of K ν .In this article we use three descriptions of Tam( ν ), as each provides a useful viewpoint.The ( I, J )-tree description shows that the length-framed triangulation is combinatoriallythe (
I, J )-Tamari complex, which is constructed using (
I, J )-trees. The ν -Dyck path per-spective makes it clear when the length-framed and planar-framed triangulations coincide(see Proposition 5.5). Finally, the ν -tree description captures the combinatorial structure ofthe triangulation succinctly, with ν -trees playing an analogous role to ν -Dyck paths in theplanar-framed triangulation (compare Figures 8 and 9). E N E N E E N E E N Figure 4.
Three corresponding ν -Catalan objects, with ν = N EN E N E .A ν -Dyck path (left). An ( I, J )-tree, with I = { , , , , , } and J = { , , , } (center). The path ν can be read from the blue labels under the( I, J )-tree. A ν -tree (right), which is a grid representation of the ( I, J )-tree inthe center.4.1.1.
The ν -Tamari lattice as the rotation lattice on ν -Dyck paths. We first give the descrip-tion of Tam( ν ) in terms of ν -Dyck paths. A valley of a lattice path is a point p at the endof an east step that is immediately followed by a north step. Let µ be a ν -Dyck path. Forany lattice point p on µ , let horiz ν ( p ) denote the maximum number of east steps that canbe added to the right of p without crossing ν . For example, horiz ν ( p ) of the lattice pointson the ν -Dyck path in Figure 4 are 0 , , , , , , , , ,
0) to (5 , ν -Dyck paths can then be endowed with the structure of a poset with the covering relation ⋖ ν defined as follows. If p is a valley of µ , let q be the first lattice point in µ after p with horiz ν ( p ) = horiz ν ( q ), and let µ [ p,q ] denote the subpath of µ between p and q . Definea rotation on µ at p by switching the east step preceding p with the subpath µ [ p,q ] . If µ ′ isthe lattice path obtained by rotating µ at p , then µ ⋖ ν µ ′ is a covering relation in Tam( ν ).Let < ν denote the transitive closure of the relation ⋖ ν . Definition 4.1 (Pr´eville-Ratelle – Viennot [15]) . The ν -Tamari lattice Tam( ν ) is a latticeon the set of ν -Dyck paths induced by the relation < ν .The leftmost lattice in Figure 5 gives an example of the ν -Tamari lattice indexed by ν -Dyckpaths. Figure 5.
The ν -Tamari lattice indexed by ν -Dyck paths (left), ( I, J )-trees(center), and ν -trees (right) for ν = N EN E N E .4.1.2. The ν -Tamari lattice as the flip lattice on ( I, J ) -trees. Next, we consider the descrip-tion of Tam( ν ) in terms of ( I, J )-trees as introduced by Ceballos, Padrol and Sarmiento [8].
Definition 4.2.
Let k ∈ Z and let I ⊔ J be a bipartition of [ k ] such that 1 ∈ I and k ∈ J .An ( I, J )- tree is a maximal subgraph of the complete bipartite graph K | I | , | J | that is(i) increasing : each arc ( i, j ) satisfies i < j ; and(ii) non-crossing : the graph does not contain arcs ( i, j ) and ( i ′ , j ′ ) with i < i ′ < j < j ′ .To a pair ( I, J ) we can associate a unique lattice path ν as follows. Assign to the i -thelement of I the label E i and to the j -th element of J the label N j . Reading the labels of thenodes k = 2 , . . . , k − ν from (0 ,
0) to ( | I |− , | J |− ν determines a unique pair ( I, J ),and hence a unique set of (
I, J )-trees. Let T ν denote the set of ( I, J )-trees determined by ν .Given two ( I, J )-trees T and T ′ in T ν , we say that T ′ is an increasing flip of T if T ′ isobtained from T by replacing an arc ( i, j ) with an arc ( i ′ , j ′ ), where i < i ′ and j < j ′ . Definea relation on T ν by T ⋖ I,J T ′ whenever T ′ is obtained from T by an increasing flip. Thetransitive closure < I,J of the relation T ⋖ I,J T ′ gives a lattice structure on T ν (see [8, Lemma3.1]). Proposition 4.3 ([8, Theorem 3.4]) . The increasing flip lattice on the set of ( I, J ) -trees in T ν is isomorphic to Tam( ν ) . UNIFYING FRAMEWORK 11
The lattice in the center of Figure 5 gives a ν -Tamari lattice with vertices indexed by( I, J )-trees. The following corollary is then immediate.
Corollary 4.4.
The Hasse diagram of the ν -Tamari lattice is the graph whose vertices arethe ( I, J ) -trees determined by ν , with edges between ( I, J ) -trees that differ by exactly one arc. The ν -Tamari lattice as the rotation lattice on ν -trees. Given a lattice path ν from(0 ,
0) to ( b, a ), let L ν denote the set of lattice points in the plane which lie weakly above ν inside the rectangle defined by (0 ,
0) and ( b, a ). An (
I, J )-tree T ∈ T ν can be represented asa point configuration in L ν . For each arc ( E x , N y ) in T , we associate the point ( x, y ) in L ν .The collection of points corresponding to the arcs of T is called the grid representation of T . These grid representations were studied in detail in [9] under the name ν -trees (see [9,Remark 3.7]). The word ‘tree’ is justified by the fact that each point except (0 , a ) in a gridrepresentation has either one point above it in the same column or one point to its left inthe same row, but not both [9, Lemma 2.2]. Thus we can connect each point to the pointabove it or to its left, forming a rooted binary tree with a root at (0 , a ). Figure 4 (right)gives an example of a ν -tree, which is the grid representation of the ( I, J )-tree in the center.The non-crossing condition for arcs in an (
I, J )-tree can be translated to ν -trees and a ν -treecan then be defined without reference to an ( I, J )-tree as follows.
Definition 4.5.
Two lattice points p and q in L ν are said to be ν -incompatible if p issouthwest or northeast of q , and the smallest rectangle containing p and q contains onlylattice points of L ν . The points p and q are ν -compatible if they are not ν -incompatible. A ν -tree is a maximal set of ν -compatible points in L ν .If a ν -tree has points p , q and r such that r is the southwest corner of the rectangledetermined by p and q (with p northwest of q or vice versa), then replacing r with the latticepoint at the point at the northeast corner of the rectangle is called a (right) rotation . Forexample, in Figure 4, the only possible rotation in the ν -tree replaces the lattice point (1 , , ν -tree are a direct translation of increasing flips for ( I, J )-trees.Define a partial order < ν on the set of ν -trees given by a covering relation T ⋖ ν T ′ if andonly if T ′ is formed from T by a rotation. This partial order is the rotation lattice of ν -trees[9, Theorem 2.8]. The rightmost lattice in Figure 5 shows the ν -Tamari lattice indexed with ν -trees. Proposition 4.6 ([9, Theorem 3.3]) . The rotation lattice on the set of ν -trees is isomorphicto Tam( ν ) . The triangulation.
In this subsection we study the length-framed triangulation of F car( ν ) and show its connection with Tam( ν ). To define the length framing of car( ν ), or anyframing for that matter, we need to be able to distinguish between the multiedges. To thatend, we label multiedges between two vertices (from top to bottom in a planar drawing ofcar( ν )) with increasing natural numbers (see Figure 3). Definition 4.7.
Let G be a graph on the vertex set { , . . . , n + 1 } . Define the length of adirected edge ( i, j ) to be j − i . Given an inner vertex i ∈ [2 , n ] of G , the length framing for G at i is the pair of linear orders ( ≺ in( i ) , ≺ out( i ) ) where longer edges precede shorter edges andmultiedges with smaller labels precede ones with larger labels. Figure 3 gives an example ofthe length framing of car( ν ) with ν = N E N EN N E N E . Recall from Section 1 that the vertices of F car( ν ) are determined by routes (unitary flows)in car( ν ). These are completely characterized by two edges in car( ν ): the initial edge thatis of the form (1 , j + 1) with label i , and the terminal edge that is of the form ( ℓ + 1 , n + 1)(which always has label i = 1) with 1 ≤ j ≤ ℓ < n . We denote such a route by R j,i,ℓ . Lemma 4.8.
The set of routes R ν in the ν -caracol graph car( ν ) is in bijection with the set A ν of possible arcs in the ( I, J ) -trees in T ν .Proof. We define a map ϕ : R ν → A ν . The elements in the sets I and J respectivelycorrespond to the N and E steps in the path ν = EνN , as in Figure 4. Describing thebijection in terms of the N and E steps is easier than using the elements of I and J , so weadd indices to the N and E steps in order to distinguish between them. First index the j -thleft-to-right N step in ν by j , then index each E with a pair ( j, i ) where j is index of thenext N j in ν , and i is the number of steps taken to reach N j . Now, arcs in the ( I, J )-treesin T ν can be expressed as pairs of the form ( E j,i , N ℓ ). Recall that the routes in car( ν ) are ofthe form R j,i,ℓ . We define the map ϕ by ϕ ( R j,i,ℓ ) = ( E j,i , N ℓ ). Figure 6 shows an example ofthis correspondence between routes and arcs.For a route R j,i,ℓ , we have that 1 ≤ j < n , and i ≤ in j +1 , and hence E j,i is the label for anode in an ( I, J )-tree in T ν . Since 1 ≤ ℓ < n , we also have that N ℓ is the label of a node.Now ( E j,i , N ℓ ) is a valid arc in A ν since j ≤ ℓ , and the map ϕ is thus well-defined. Definethe inverse map ϕ − : A ν → R ν by ϕ − (( E j,i , N ℓ )) = R j,i,ℓ . Since ( E j,i , N ℓ ) is an arc, thenode N j +1 is preceded by at least i many nodes labeled by E steps. Thus car( ν ) has an edge(1 , j + 1) labeled i . Now R j,i,ℓ is the unique route in R ν determined by j, i and ℓ , and so ϕ − is well-defined. It is clear that ϕ ◦ ϕ − and ϕ − ◦ ϕ are identity maps. (cid:3) E , E , E , E , E , E , N N N N
123 45 6 7 8 9
Figure 6.
A maximal clique of routes (left) representing a simplex in thelength-framed triangulation of F car( ν ) for ν = N EN E N E . The bijection ϕ of Lemma 4.8 sends each route to the corresponding arc of the ( I, J )-tree onthe right. The bijection Φ in the proof of Theorem 1.2 sends the maximalclique to the (
I, J )-tree.
Lemma 4.9.
Let ≺ denote the length framing, and let ϕ be the bijection in Lemma 4.8.Two routes R j,i,ℓ and R j ′ ,i ′ ,ℓ ′ in the framed graph (car( ν ) , ≺ ) are coherent if and only if ϕ ( R j,i,ℓ ) = ( E j,i , N ℓ ) and ϕ ( R j ′ ,i ′ ,ℓ ′ ) = ( E j ′ ,i ′ , N ℓ ′ ) are non-crossing arcs in A ν .Proof. We can assume without loss of generality that ℓ < ℓ ′ (if ℓ = ℓ ′ , the arcs are non-crossing and the routes are coherent). There are two ways in which the arcs ϕ ( R j,i,ℓ ) and ϕ ( R j ′ ,i ′ ,ℓ ′ ) can cross: (1) when j < j ′ , or (2) when j = j ′ with i ′ < i . In both cases we havethat R j,i,ℓ and R j ′ ,i ′ ,ℓ ′ are incoherent at the vertex j ′ (see Figure 7). Conversely, if R j,i,ℓ and R j ′ ,i ′ ,ℓ ′ are incoherent, they must be incoherent at a minimal vertex j ′ . Thus either j < j ′ UNIFYING FRAMEWORK 13 or j = j ′ with i ′ < i , which are precisely the ways in which ϕ ( R j,i,ℓ ) and ϕ ( R j ′ ,i ′ ,ℓ ′ ) cancross. (cid:3) Case 1 : R j,i,ℓ j ℓ n + 1 i R j ′ ,i ′ ,ℓ ′ j ′ ℓ ′ n + 1 i ′ E j,i N ℓ E j ′ ,i ′ N ℓ ′ ϕ ( R j,i,ℓ ) ϕ ( R j ′ ,i ′ ,ℓ ′ ) −→←− Case 2 : R j,i,ℓ j = j ′ ℓ n + 1 i R j ′ ,i ′ ,ℓ ′ j = j ′ ℓ ′ n + 1 i ′ < i E j,i N ℓ E j ′ ,i ′ N ℓ ′ ϕ ( R j,i,ℓ ) ϕ ( R j ′ ,i ′ ,ℓ ′ ) −→←− Figure 7.
The two cases in the proof of Lemma 4.9.
Theorem 1.2.
The length-framed triangulation of F car( ν ) is a regular unimodular triangu-lation whose dual graph is the Hasse diagram of the ν -Tamari lattice Tam( ν ) .Proof. By Lemma 4.9, the bijection ϕ in Lemma 4.8 extends to a bijection Φ from the set ofmaximal cliques of routes in the length-framed car( ν ) to the set T ν of ( I, J )-trees determinedby ν . Two simplices in a DKK triangulation of a flow polytope are adjacent if and only ifthey differ by a single vertex, that is, if the corresponding maximal cliques differ by a singleroute. Under the bijection Φ, two simplices are adjacent if and only if their corresponding( I, J )-trees differ by a single arc, which by Corollary 4.4 is precisely the description of theHasse diagram of the ν -Tamari lattice. (cid:3) Example 4.10.
Let ν = N EN E N E . One example of the bijection Φ between cliquesof routes of car( ν ) and ( I, J )-trees is illustrated in Figure 6. The dual graph of the length-framed triangulation of F car( ν ) is shown in Figure 5.In [8] Ceballos, Padrol, and Sarmiento introduced the ( I, J ) -Tamari complex A I,J as theflag simplicial complex on { ( i, j ) ∈ I × J | i < j } whose minimal non-faces are the pairs { ( i, j ) , ( i ′ , j ′ ) } with i < i ′ < j < j ′ , that is, the complex on collections of non-crossing arcsof ( I, J )-trees. The following result is then a corollary of Theorem 1.2.
Corollary 4.11.
Let ν be the lattice path from (0 , to ( b, a ) associated to the pair ( I, J ) .The length-framed triangulation of F car( ν ) is a geometric realization of the ( I, J ) -Tamaricomplex of dimension a + b in R a + b +3 . Remark 4.12.
A graph closely related to car( ν ) can produce a realization of the ( I, J )-Tamari complex in R a + b +1 that is a simple projection of the geometric realization of Corollary4.11. The geometric realization of Corollary 4.11 is integrally equivalent to the first of threerealizations given in [8, Theorem 1.1]. A second description in terms of ν -trees. We conclude this section by indexingsimplices in the length-framed triangulation of F car( ν ) by ν -trees, which in terms of thelattice path ν is an analogous counterpart to the ν -Dyck path description of the planar-framed triangulation in Section 5. The vertices (routes) of F car( ν ) are encoded by the latticepoints in L ν , with the lattice points in each ν -tree corresponding to a maximal simplex in thelength-framed triangulation. Furthermore, it then follows that two simplices are adjacent inthe length-framed triangulation if their corresponding ν -trees differ by a rotation. Lemma 4.13.
Let ν be a lattice path from (0 , to ( b, a ) . There exists a (first) bijection θ : R ν → L ν between the set R ν of routes in car( ν ) and the set L ν of lattice points in therectangle defined by (0 , and ( b, a ) that lie weakly above ν .Proof. The map ϕ : R ν → A ν in Lemma 4.8 gives a bijection. Let γ : A ν → L ν be givenby ( E x , N y ) ( x, y ). For any arc ( E x , N y ), since E x appears before N y in ν , we have γ ( E x , N y ) = ( x, y ) ∈ L ν . The inverse γ − is well-defined since if ( x, y ) ∈ L ν , then N y ispreceded by at least x E steps in ν , and hence ( E x , N y ) ∈ A ν . Now θ := γ ◦ ϕ is the desiredbijection. (cid:3) A second bijection between the routes R ν of car( ν ) and the lattice points L ν is given inLemma 5.2. The present bijection θ leads to a characterization of the routes which appearin every simplex of the length-framed triangulation of F car( ν ) .Recall from Lemma 4.9 that two routes in R ν are coherent if and only if their correspondingarcs in A ν are non-crossing. Since the non-crossing condition for arcs in ( I, J )-trees translatesto the ν -compatibility condition of ν -trees, we have that routes in R ν are coherent if and onlyif their corresponding lattice points via θ are ν -compatible. If we associate each lattice pointin L ν with the corresponding route in car( ν ) via the bijection θ in Lemma 4.13, then thelattice points in a ν -tree correspond to a maximal clique of routes. Two adjacent simplicesin the length-framed triangulation of F car( ν ) differ by a single vertex, and the corresponding ν -trees differ by a single lattice point via a rotation. Note that a ν -tree will always containthe root (0 , a ), the valleys of the lattice path ν , along with each initial point of any initial N steps of ν and each terminal point of any terminal E steps of ν . These points correspond tothe routes which are coherent with all other routes in the length-framing, and thus appearin every simplex of the length-framed triangulation.In the example in Figure 8, the routes which appear in every simplex of the length-framedtriangulation of F car( ν ) are labeled 1 , , , , Figure 8.
A maximal clique of routes (left) representing a simplex in thelength-framed triangulation of F car( ν ) for ν = N EN E N E . The bijection θ of Lemma 4.13 sends each route to a lattice point in the corresponding ν -tree(right). UNIFYING FRAMEWORK 15 The planar-framed triangulation
The goal of this section is to show that the flow polytope F car( ν ) has a regular unimodulartriangulation whose dual graph structure is given by the Hasse diagram of a principal orderideal I ( ν ) in Young’s lattice. A consequence of this is we can construct a family of posets Q ν so that the dual graph structure of the canonical triangulation of the order polytope O ( Q ν )is also I ( ν ).5.1. Principal order ideals in Young’s lattice.
Recall that
Young’s lattice Y is the poseton integer partitions λ with covering relations λ ⋖ λ ′ if λ is obtained from λ ′ by removingone corner box of λ ′ . Note that a lattice path ν in the rectangular grid defined by (0 ,
0) to( b, a ) defines a partition λ ( ν ) = ( λ , . . . , λ a ) by letting λ k = b − P ai = a − k +1 ν i for k = 1 , . . . , a .The Young diagram for λ ( ν ) may be visualized as the region within the rectangle from (0 , b, a ) which lies NW of ν . For example in Figure 1, ν = N E N EN N E N E defines thepartition λ ( ν ) = (6 , , , order ideal of a poset P is a subset I ⊆ P with the propertythat if x ∈ I and y ≤ x , then y ∈ I . An ideal is said to be principal if it has a single maximalelement x ∈ P , and such an ideal will be denoted by I ( x ).If µ is a ν -Dyck path, then it lies weakly above the path ν and so µ can be identifiedwith a partition λ ( µ ) that is contained in λ ( ν ). Thus there is a one-to-one correspondencebetween the set of ν -Dyck paths with the set of elements in the order ideal I ( ν ) := I ( λ ( ν ))in Y . Under this correspondence, in terms of ν -Dyck paths, a path π covers a path µ if andonly if π can be obtained from µ by replacing a consecutive N E pair by a EN pair. See theright side of Figure 10 for an example of I ( ν ) with ν = N EN E N E .5.2. The triangulation.Definition 5.1.
Let G be a planar graph that affords a planar embedding in the plane suchthat if vertex i is at the coordinates ( x i , y i ), then x i < x j for all i < j . This leads to naturalorderings ( ≺ in( i ) , ≺ out( i ) ) at every inner vertex i of G as follows: with respect to the planarembedding of G , the incoming edges at the vertex i are ordered in increasing order from thetop to the bottom, and the same for the outgoing edges from the vertex i . This is the planarframing for G .It is clear that the graphs car( ν ) have a planar embedding with the properties of Defini-tion 5.1 if it is embedded so that the path 1 , . . . , n + 1 lies on the x -axis. Figure 3 gives anexample of the planar framing of car( ν ) with ν = N E N EN N E N E . Lemma 5.2.
Let ν be a lattice path from (0 , to ( b, a ) . There exists a (second) bijection ψ : R ν → L ν between the set R ν of routes in car( ν ) and the set L ν of lattice points in therectangle defined by (0 , and ( b, a ) that lie weakly above ν .Proof. We fix an embedding of car( ν ) in the plane so that the path 1 , , . . . , n + 1 lies on the x -axis. Define a map ψ : R ν → L ν by ψ ( R ) = ( j, ℓ ), where j is the number of bounded facesof car( ν ) that lie below R and above the x -axis, and ℓ is the number of bounded faces thatlie below R and the x -axis. See Figure 9 for an example.To see that ψ is well-defined, first note that any planar embedding of car( ν ) has m − n = a + b bounded faces, by Euler’s formula. In particular, the fixed planar embedding of car( ν ) has a bounded faces below the x -axis and b bounded faces above the x -axis, so 0 ≤ j ≤ b and0 ≤ ℓ ≤ a . To show that the lattice point ψ ( R ) = ( j, ℓ ) lies weakly above ν , we must showthat 0 ≤ j ≤ ν + · · · + ν ℓ for a fixed 0 ≤ ℓ ≤ a . If ψ ( R ) = ( j, ℓ ), this means that the last edgeof the route R is ( ℓ + 2 , n + 1). By counting the in-degrees of the vertices 3 , , . . . , ℓ + 2 incar( ν ), we see that there are at most ν + · · · + ν ℓ edges in car( ν ) of the form (1 , j ) embeddedabove the x -axis for 3 ≤ j ≤ ℓ + 2. Consequently, there are at most ν + · · · + ν ℓ boundedregions which can lie below the route R and above the x -axis. Therefore, ψ is well-defined.To see that ψ is invertible, let ( j, ℓ ) ∈ L ν so that 0 ≤ j ≤ ν + · · · + ν ℓ . Then there existsa unique 1 < k ≤ ℓ + 2 such that there are j bounded faces of the embedded car( ν ) whichlie between the edge (1 , k ) and the x -axis. Let R be the route whose first edge is (1 , k ) andlast edge is ( ℓ + 2 , n + 1) (recall from Section 4 that every route in car( ν ) is completelycharacterized by these two edges). Then ψ ( R ) = ( j, ℓ ) as required, and therefore, ψ is abijection. (cid:3) The above bijection ψ leads to a characterization of the routes which appear in everysimplex of the planar-framed triangulation of F car( ν ) . As we will see in Theorem 1.3, thebijection ψ extends to a bijection Ψ in which a maximal clique of routes in the planarframing of car( ν ) correspond to the collection of lattice points in a ν -Dyck path. A ν -Dyckpath always contains the lattice points of any initial N steps of ν and any terminal E stepsof ν . Hence, under this bijection, these points correspond to the routes which are coherentwith all other routes, and thus appear in every simplex of the planar-framed triangulation.For example in Figure 9, the routes which appear in every simplex of the planar-framedtriangulation of F car( ν ) are labeled 1 , , , Figure 9.
A maximal clique of routes (left) representing a simplex in theplanar-framed triangulation of F car( ν ) for ν = N EN E N E . The extension Ψof the bijection ψ of Lemma 5.2 sends this clique to the ν -Dyck path on theright.Given two lattice points ( x , y ) and ( x , y ) with x < x are said to be incompatible if y > y . Otherwise, any other pair of lattice points are said to be compatible . Maximal setsof compatible lattice points lying above ν determine a unique ν -Dyck path. Lemma 5.3.
Let ≺ denote the planar framing, and let ψ be the bijection in Lemma 5.2.Two routes R and R in the framed graph (car( ν ) , ≺ ) are coherent if and only if ψ ( R ) and ψ ( R ) are compatible.Proof. A result of M´esz´aros, Morales and Striker [14, Lemma 6.5] states that two routes ina planar framing of a graph G are coherent if and only if they are non-crossing in G . Let R and R be two routes in car( ν ) and ψ ( R ) = ( j , ℓ ), ψ ( R ) = ( j , ℓ ). UNIFYING FRAMEWORK 17
Suppose R and R are coherent with ℓ < ℓ . Then the fact that R and R are non-crossing implies that j ≤ j , hence ( j , ℓ ) and ( j , ℓ ) are compatible. On the other handsuppose R and R are not coherent. Without loss of generality, we may assume that ℓ < ℓ (for otherwise, if ℓ = ℓ then the routes are coherent). Let k be the smallest vertex at whichthe routes cross. Then j ≤ j ≤ k , and hence ( j , ℓ ) and ( j , ℓ ) are not compatible. (cid:3) Theorem 1.3.
The planar-framed triangulation of F car( ν ) is a regular unimodular triangu-lation whose dual graph is the Hasse diagram of the principal order ideal I ( ν ) in Young’slattice Y .Proof. By Lemma 5.3, the bijection ψ in Lemma 5.2 extends to a bijection Ψ from maximalcliques of routes in the planar-framed car( ν ) to maximal sets of compatible lattice pointslying above ν , which are ν -Dyck paths. Two simplices in a DKK triangulation of a flowpolytope are adjacent if and only if they differ by a single vertex. Under the bijection Ψ, twosimplices are adjacent if and only if their corresponding ν -Dyck paths π and π differ by asingle lattice point. Let ( x , y ) ∈ π and ( x , y ) ∈ π be the lattice points which are notcontained in both paths. Assume without loss of generality that x < x . Since these latticepoints are not compatible, we must have y > y . Thus ( x , y ) is in the top left corner ofthe single square determined by ( x , y ) and ( x , y ), while ( x , y ) is in the bottom left. Inother words, π and π differ by a transposition of a consecutive N E pair, which is preciselythe description of the covering relation in the principal order ideal I ( ν ). (cid:3) Example 5.4.
Let ν = N EN E N E . The bijection Ψ between cliques of routes of car( ν )and ν -Dyck paths is shown in Figure 9. The dual graph of the planar-framed triangulationof F car( ν ) is shown in Figure 10 on the right. Figure 10.
The ν -Tamari lattice (left) and the Hasse diagram of the orderideal I ( ν ) ⊆ Y (right) for ν = N EN E N E . These are the dual graphs of thelength-framed and planar-framed triangulations of F car( ν ) .5.3. Comparing the length-framed and planar-framed triangulations.
A specialcase when the dual structure of the length-framed and planar-framed triangulations of F car( ν ) are the same is given by the following proposition. Proposition 5.5.
When ν = E a N b , so that the set of ν -Dyck paths is the set of all latticepaths from (0 , to ( b, a ) , the length-framed triangulation and the planar-framed triangulationof F car( ν ) have the same dual structure. Proof.
We use the ν -Dyck path description (see Section 4.1) of the ν -Tamari lattice Tam( ν )in this proof. Let µ be a ν -Dyck path. For any valley point p of µ , the next lattice point q in µ with horiz ν ( p ) = horiz ν ( q ) is the next lattice point after p . This is because thehorizontal distance of any of the lattice points in a run of consecutive N steps is the samewhen ν = E a N b . Performing a rotation on µ at the valley point p to obtain the ν -Dyckpath µ ′ is then the same as exchanging the EN pair centered at p with an N E pair ofsteps in µ . Thus µ < ν µ ′ is a covering relation in the lattice Tam( ν ) if and only if it is acovering relation in the dual order ideal I ( ν ) ∗ . Therefore, Tam( ν ) = I ( ν ) ∗ . Lastly, since thepartition λ ( ν ) = ( b a ) is rectangular, then I ( ν ) is self-dual. Therefore, Tam( ν ) and I ( ν ) areisomorphic. (cid:3) A connection with order polytopes.
In this subsection, G is a planar graph on thevertex set [ n +1] with a planar embedding satisfying the properties outlined in Definition 5.1.We further assume that the in-degree and out-degree of each vertex i for i = 2 , . . . , n is atleast one. A result of M´esz´aros, Morales and Striker [14, Theorem 3.11] states that for sucha graph G , the flow polytope F G is integrally equivalent to the order polytope O ( P G ), where P G is a poset that is induced by the bounded faces of the planar embedding of G .In this section, we explain how our results for flow polytopes on the caracol graphs car( ν )lead to analogous results for a certain class of order polytopes O ( Q ν ). We give a brief back-ground of known results relating order polytopes and flow polytopes following the expositionof [14], and explain their implications when applied to car( ν ).Let ( P, ≤ P ) be a finite poset with elements { p , . . . , p d } . The order polytope of P is theset of points O ( P ) = (cid:8) ( x p , . . . , x p d ) ∈ [0 , d | x p i ≤ x p j if p i ≤ P p j (cid:9) . Given a linear extension σ : P → [ d ] of the poset P , i.e. an order preserving bijection with[ d ] endowed with its natural order, define the simplex∆ σ = (cid:8) ( x p , . . . , x p d ) ∈ [0 , d | x σ − (1) ≤ · · · ≤ x σ − ( d ) (cid:9) . The canonical triangulation of O ( P ), first defined by Stanley [16], is the set of simplices { ∆ σ | σ is a linear extension of P } . Thus the normalized volume of O ( P ) is the number of linear extensions of P .For a planar graph G with a fixed embedding in the plane, the truncated dual graph G ∗ of G is the dual graph whose vertices correspond to the bounded faces of G . Viewing G ∗ asembedded on the plane also, then the orientation on the edges of G induces an orientationon the edges of G ∗ . The graph G ∗ then induces the Hasse diagram of a poset that is denotedby P G . See Figure 11.In the case ν = N E ν · · · N E ν a is a lattice path from (0 ,
0) to ( b, a ), then the poset P car( ν ) has a + b elements (corresponding to the bounded faces of the embedded car( ν )), and isconstructed in the following way. There are a elements labeled N , . . . , N a corresponding tothe bounded faces that are uniquely determined by each pair of edges of the form ( j +2 , n +1)and ( j + 3 , n + 1) for j = 1 , . . . , a . For each j = 1 , . . . , a , if there are k edges of the form(1 , j + 2) in car( ν ), then there are k elements labeled E j, , . . . , E j,k . Since there are b edgesof the form (1 , j + 2) for j = 1 , . . . , a , then there are b elements labeled with an E . The UNIFYING FRAMEWORK 191 2 3 4 5 6 7 8 E E E E E E E N N N N N Figure 11.
Let ν = N E N EN N E N E . The truncated dual graph of car( ν )is the graph whose vertices are the bounded faces of the embedded car( ν ) (left).The Hasse diagram of the poset Q ν = P car( ν ) is induced by the truncated dual(right).relations in P car( ν ) are N i < N j for 1 ≤ i < j ≤ a , E j,k < E ℓ,n if ( j, k ) appears before ( ℓ, n )in lexicographic order, and N i < E j,k if i ≤ j .Note that if we think of N i as N i, , then listing the subscripts of these N s and E s lexico-graphically recovers the ν -Dyck path. With this observation, it means that we can define aclass of posets Q ν (equal to P car( ν ) ) indexed by lattice paths ν without any reference to flowpolytopes. See Figure 11.It was first observed by Postnikov (also see M´esz´aros, Morales and Striker [14, Theorem1.3]) that the canonical triangulation of O ( P G ) is the same as the planar-framed DKKtriangulation of F G up to an integral equivalence. Combined with Theorems 1.2 and 1.3, wehave the following two corollaries. Corollary 5.6.
The canonical triangulation of the order polytope O ( Q ν ) has dual graphwhich is the Hasse diagram of the principal order ideal I ( ν ) in Young’s lattice. Corollary 5.7.
The order polytope O ( Q ν ) has a regular unimodular triangulation whosedual graph is the ν -Tamari lattice Tam( ν ) . For a poset P , let J ( P ) denote the lattice of order ideals of P ordered by inclusion.From [16, Section 5], the maximal chains of J ( P ) are in bijection with the simplices in thecanonical triangulation of O ( P ). This gives another perspective on the direct relationshipbetween simplices in the planar-framed triangulation of F car( ν ) , maximal cliques in the flowpolytope F car( ν ) , ν -Dyck paths, maximal chains in J ( Q ν ), and simplices in the canonicaltriangulation of O ( Q ν ). In this case, the Hasse diagram of J ( Q ν ) can be obtained by takingthe lattice on the points L ν which lie above ν , and rotating it counterclockwise by 45 degrees.Having obtained results for order polytopes via methods for flow polytopes, we now endthis section with a result for flow polytopes via methods for order polytopes. Corollary 5.8.
Let ν = N E ν · · · N E ν a be a lattice path from (0 , to ( b, a ) . Let peak( ν ) denote the number of consecutive N E pairs in ν . The number of facets of F car( ν ) is a + b +peak( ν ) .Proof. A result of Stanley [16] states that the facets of an order polytope O ( P ) correspondto the covering relations of the poset b P = P ∪ { ˆ0 , ˆ1 } . The poset b P car( ν ) consists of a lower chain ˆ0 ⋖ N ⋖ · · · ⋖ N a with a covering relations, an upper chain E ⋖ · · · ⋖ ˆ1 with b coveringrelations, and additional covering relations of the form N i ⋖ E i,k if and only if N i E i,k is aconsecutive N E pair (in other words, a peak) in ν . The result then follows since O ( P car( ν ) )is integrally equivalent to F car( ν ) . (cid:3) The h ∗ -vector of the ν -caracol flow polytope The h ∗ -vector of a lattice polytope coincides with the h -vector of any of its unimodulartriangulations [4, Theorem 10.3], so we will compute the h -vector of the planar-framedtriangulation of F car( ν ) . This extends a result of M´esz´aros [12] for the classical case when ν = (1 n ).We begin by recalling some relevant definitions from [19]. Given a simplicial complex, a shelling is an ordering F , ..., F s of its facets such that for every i < j there is some k < j such that the intersection F i ∩ F j ⊆ F k ∩ F j , and F k ∩ F j is a facet of F j . A simplicial complexis said to be shellable if it admits a shelling. The h -vectors of shellable simplicial complexeshave non-negative entries which can be computed combinatorially from the shelling order asfollows. For a fixed shelling order F , ..., F s define the restriction R j of the facet F j as the set R j := { v ∈ F j : v is a vertex in F j and F j \ v ⊆ F i for some 1 ≤ i < j } . Then the i -entry ofthe h -vector is given by h i = |{ j : | R j | = i, ≤ j ≤ s }| . Lemma 6.1.
Let C be the planar-framed triangulation of F car( ν ) interpreted as a simplicialcomplex. Any linear extension of I ( ν ) gives a shelling order of C .Proof. By Theorem 1.3 we can give the dual graph of C the structure of I ( ν ), identifyingeach facet in C with the associated ν -Dyck path in I ( ν ). For a linear extension L of I ( ν ), wecan order the facets F , ..., F s of C according to L . Let π i and π j be two ν -Dyck paths in L ,with i < j . Let π s be the minimal ν -Dyck path that covers both π i and π j , i.e. π s = π i ∨ π j in I ( ν ). Now π s contains the lattice points in π i ∩ π j , and so F i ∩ F j ⊆ F s . It is clear thatthere exists a sequence of ν -Dyck paths π s , π s , ..., π j such that each path contains the latticepoints π i ∩ π j , and each path is formed from the previous path by replacing a consecutive N E pair with EN . Given such a sequence of paths, let π k be the second to last path in thesequence. Replacing a consecutive N E pair with EN in π k yield π j . Now k < j , and F i ∩ F j is contained in every facet F s ℓ for 1 ≤ ℓ ≤ k . In particular, F i ∩ F j ⊆ F k ∩ F j . Furthermore, π k and π j differ by a single lattice point, so F k ∩ F j is a facet of F j . (cid:3) Let ν be a lattice path from (0 ,
0) to ( b, a ). For i = 0 , . . . , a , the ν -Narayana number Nar ν ( i ) is the number of ν -Dyck paths with i valleys (recall that a valley is a consecutive EN pair). The ν -Narayana polynomial is N ν ( x ) = P i ≥ Nar ν ( i ) x i . For more on thesedefinitions, see [5] or [8], for example. Theorem 1.4.
The h ∗ -polynomial of F car( ν ) is the ν -Narayana polynomial.Proof. It suffices to find the h -vector of the planar-framed triangulation of F car( ν ) . Any linearextension of the order ideal I ( ν ) gives a shelling order of the planar-framed triangulation of EFERENCES 21 F car( ν ) . The i -th entry of the h -vector is h i = |{ j : | R j | = i, ≤ j ≤ s }| = |{ paths in I ( ν ) that cover exactly i paths }| = |{ paths with exactly i valleys }| = Nar ν ( i ) . (cid:3) A different proof of Theorem 1.4 can be obtained by computing the h -vector of the length-framed triangulation of F car( ν ) , which by Corollary 4.11 is combinatorially equivalent to the( I, J )-Tamari complex with the pair (
I, J ) associated to ν , which we also call the ( I, J ) -Tamari complex . In [8, Lemma 4.5] a shelling order on facets of this complex was used toshow that the h -vector of the ( I, J )-Tamari complex is given by the ν -Narayana numbers.Since any lattice unimodular triangulation can be used to calculate the h ∗ -vector of F car( ν ) ,Theorem 1.4 provides a new proof that the h -vector of the ( I, J )-Tamari complex is givenby the ν -Narayana numbers. Acknowledgments
The second and fourth authors are extremely grateful to AIM and the SQuaRE group“Computing volumes and lattice points of flow polytopes” as some of the ideas of this workcame from discussions within the group. In particular, we want to thank Alejandro Moralesfor the many enlightening discussions and explanations on triangulations of flow polytopes.Martha Yip is partially supported by Simons Collaboration Grant 429920.
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Department of Mathematics, University of Kentucky,
Email address : [email protected] (Gonz´alez D’Le´on) Escuela de Ciencias Exactas e Ingenier´ıa, Universidad Sergio Arboleda,
Email address : [email protected] (Mayorga Cetina) Escuela de Ciencias Exactas e Ingenier´ıa, Universidad Sergio Arboleda,
Email address : [email protected] (Yip) Department of Mathematics, University of Kentucky,
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