aa r X i v : . [ m a t h . C O ] O c t ON THE H -TRIANGLE OF GENERALISED NONNESTINGPARTITIONS MARKO THIEL
Abstract.
With a crystallographic root system Φ and a positive integer k ,there are associated two Fuß-Catalan objects, the set of k -generalised nonnest-ing partitions NN ( k ) (Φ), and the generalised cluster complex ∆ ( k ) (Φ). Thesepossess a number of enumerative coincidences, many of which are capturedin a surprising identity, first conjectured by Chapoton for k = 1 and latergeneralised to k > NN ( k ) (Φ) along the way, including anearlier conjecture by Fomin and Reading giving a refined enumeration by Fuß-Narayana numbers. Introduction
For a crystallographic root system Φ, there are three well-known Coxeter-Catalanobjects [Arm09]: the set of noncrossing partitions
N C (Φ), the set of nonnestingpartitions
N N (Φ) and the cluster complex ∆(Φ). The former two and the set offacets of the latter are all counted by the same numbers, the Coxeter-Catalan num-bers
Cat (Φ). For the root system of type A n − , these reduce to objects countedby the classical Catalan numbers C n = n +1 (cid:0) nn (cid:1) , namely the set of noncrossingpartitions of [ n ] = { , , . . . , n } , the set of nonnesting partitions of [ n ] and the setof triangulations of a convex ( n + 2)-gon, respectively.Each of these Coxeter-Catalan objects has a generalisation [Arm09], a Fuß-Catalanobject defined for each positive integer k . These are the set of k -divisible noncross-ing partitions N C ( k ) (Φ), the set of k -generalised nonnesting partitions N N ( k ) (Φ)and the generalised cluster complex ∆ ( k ) (Φ). They specialise to the correspondingCoxeter-Catalan objects when k = 1. The former two and the set of facets of thelatter are counted by Fuß-Catalan numbers Cat ( k ) (Φ), which specialise to the clas-sical Fuß-Catalan numbers C ( k ) n = kn +1 (cid:0) ( k +1) nn (cid:1) in type A n − .The enumerative coincidences do not end here. Chapoton defined the M -triangle,the H -triangle and the F -triangle, which are polynomials in two variables thatencode refined enumerative information on N C (Φ),
N N (Φ) and ∆(Φ) respec-tively [Cha04, Cha06]. This allowed him to formulate the M = F conjecture[Cha04, Conjecture 1] and the H = F conjecture [Cha06, Conjecture 6.1] relatingthese polynomials through invertible transformations of variables. These conjec-tures were later generalised to the corresponding Fuß-Catalan objects by Armstrong[Arm09, Conjecture 5.3.2.]. The M = F conjecture was first proven by Athanasiadis[Ath07] for k = 1, and later by Krattenthaler [Kra06a, Kra06b] and Tzanaki [Tza08]for k >
1. We establish the H = F conjecture in the generalised form due to Arm-strong, obtaining some structural and enumerative results on N N ( k ) (Φ) along theway, including a refined enumeration by Fuß-Narayana numbers conjectured byFomin and Reading [FR05, Conjecture 10.1.]. Definitions and the Main Result
Let Φ = Φ( S ) be a crystallographic root system with a simple system S . ThenΦ = Φ + ⊔ − Φ + is the disjoint union of the set of positive roots Φ + and the setof negative roots − Φ + . Every positive root can be written uniquely as a linearcombination of the simple roots and all coefficients of this linear combination arenonnegative integers. For further background on root systems, see [Hum90]. Definethe root order on Φ + by β ≥ α if and only if β − α ∈ h S i N ,that is, β ≥ α if and only if β − α can be written as a linear combination of simpleroots with nonnegative integer coefficients. The set of positive roots Φ + with thispartial order is called the root poset . An order filter in this poset is a subset I ofΦ + such that whenever α ∈ I and β ≥ α , then also β ∈ I . The support of a root β ∈ Φ + is the set of all α ∈ S with α ≤ β . For α ∈ S , define I ( α ) as the order filtergenerated by α , that is the set of all roots β ∈ Φ + that have α in their support.We call a root system classical if all its irreducible components are of type A , B , C or D . Otherwise we call it exceptional .Let V = h Φ i = h S i be the ambient vector space spanned by the roots. Definethe k -Catalan arrangement as the hyperplane arrangement in V given by the hy-perplanes H α,i = { x ∈ V | h x, α i = i } for α ∈ Φ and i ∈ { , , . . . , k } . Theconnected components of V \ S α,i H α,i are called the regions of the arrangement,and those regions R with h x, α i ≥ x ∈ R and all α ∈ Φ + are called dominant . A supporting hyperplane of a dominant region R is called a wall of R ,it is a separating wall or floor of R if the origin and R lie on different sides ofthe hyperplane. It is a ceiling if R and the origin lie on the same side of the hy-perplane. A hyperplane is called i -coloured if it is of the form H α,i for some α ∈ Φ + .Now let k be a positive integer and consider a descending (multi)chain of orderfilters: a tuple I = ( I , I , . . . , I k ) with I ⊇ I ⊇ . . . ⊇ I k . This chain is called geometric if the following conditions hold:( I i + I j ) ∩ Φ + ⊆ I i + j for all i, j ∈ { , , . . . , k } and( J i + J j ) ∩ Φ + ⊆ J i + j for all i, j ∈ { , , . . . , k } with i + j ≤ k ,where I = Φ + , J i = Φ + \ I i for i ∈ { , , . . . , k } , and I i = I k for i > k . The set N N ( k ) (Φ) of k -generalised nonnesting partitions of Φ is the set of all geometricchains of k order filters in the root poset of Φ.For I ∈ N N ( k ) (Φ), and α ∈ Φ + , define k α ( I ) = max { k + k + . . . + k r | α = α + α + . . . + α r and α i ∈ I k i for all i } .A positive root α is called a rank l indecomposable element [Ath05, Definition 3.8.]of a k -nonnesting partition I if α ∈ I l , k α ( I ) = l , α / ∈ I i + I j for all i, j ∈ { , , . . . , k } with i + j = l , and for all β ∈ Φ + , if α + β ∈ I t for some t ≤ m and k α + β ( I ) = t ,then β ∈ I t − l . A rank l indecomposable element α is called a rank l simple element of I if α ∈ S . The rank k indecomposable elements of I are precisely the elementsof I k that are not in I i + I j for all i, j ∈ { , , . . . , k } with i + j = k [Ath05, Lemma3.9.]. Notice that all α ∈ I k ∩ S are automatically rank k indecomposable.Now we can define the H -triangle [Arm09, Definition 5.3.1.] as H k Φ ( x, y ) = X I ∈ NN ( k ) (Φ) x | i ( I ) | y | s ( I ) | , N THE H -TRIANGLE OF GENERALISED NONNESTING PARTITIONS 3 where i ( I ) is the set of rank k indecomposable elements of I and s ( I ) is the set ofrank k simple elements of I .There is a natural bijection φ [Ath05, Theorem 3.6.] from the set of dominantregions R of the k -Catalan arrangement of Φ to N N ( k ) (Φ), such that H α,i is a floorof R if and only if α is a rank i indecomposable element of φ ( R ).Let Φ ( k ) ≥− be the set of k-coloured almost positive roots of Φ, containing oneuncoloured copy of each negative simple root and k copies of each positive root,each with a different colour from the colour set { , , . . . , k } . Then there exists asymmetric binary relation called compatibility [FR05, Definition 3.1.] on Φ ( k ) ≥− suchthat all uncoloured negative simple roots are pairwise compatible and for α ∈ S anuncoloured simple root and β ( i ) ∈ Φ + a positive root with colour i , − α is compati-ble with β ( i ) if and only if α is not in the support of β . Notice that the colour i of β ( i ) does not matter in this case.Define a simplicial complex ∆ ( k ) (Φ) as the set of all subsets A ⊆ Φ ( k ) ≥− suchthat all k -coloured almost positive roots in A are pairwise compatible. This isthe generalised cluster complex of Φ. This simplicial complex is pure , all facetshave cardinality n , where n = | S | is the rank of Φ. The subcomplex consisting ofthose faces containing no negative simple root is called the positive part of ∆ ( k ) (Φ).Now we can define the F -triangle [Arm09, Definition 5.3.1.] as F k Φ ( x, y ) = X A ∈ ∆ ( k ) (Φ) x | A + | y | A − | = X l,m f kl,m (Φ) x l y m ,where A + is the set of coloured positive roots in A and A − is the set of uncolourednegative simple roots in A . Thus f kl,m (Φ) is the number of faces of ∆ ( k ) (Φ) contain-ing exactly l coloured positive roots and exactly m uncoloured negative simple roots.Consider the Weyl group W = W (Φ) of the root system Φ. A Coxeter element in this group is a product of all the simple reflections in some order. Let T denotethe set of reflections in W . For w ∈ W , define the absolute length l T ( w ) of w as theminimal l such that w = t t · · · t l for some t , t , . . . , t l ∈ T . Define the absoluteorder on W by u ≤ T v if and only if l T ( u ) + l T ( u − v ) = l T ( v ).Fix a Coxeter element c ∈ W . A k -delta sequence is a sequence δ = ( δ , δ , . . . , δ k )with δ i ∈ W for all i ∈ { , , . . . , k } such that c = δ δ · · · δ k and l T ( c ) = P ki =0 l T ( δ i ).Define a partial order on k -delta sequences by δ ≤ ǫ if and only if δ i ≤ T ǫ i for all i ∈ { , , . . . , k } .The set of k -delta sequences with this partial order is called the poset of k -divisiblenoncrossing partitions N C ( k ) (Φ) [Arm09, Definition 3.3.1.]. We drop the choice ofthe Coxeter element c from the notation, since a different choice of Coxeter elementresults in a different but isomorphic poset.Now we can define the M -triangle [Arm09, Definition 5.3.1] as M k Φ ( x, y ) = X δ,ǫ ∈ NC ( k ) (Φ) µ ( δ, ǫ ) x n − rk ( ǫ ) y n − rk ( δ ) ,where rk is the rank function of the graded poset N C ( k ) (Φ), µ is its M¨obius func-tion, and n is the rank of Φ. MARKO THIEL
As mentioned in the introduction,
N C ( k ) (Φ), N N ( k ) (Φ) and the set of facets of∆ ( k ) (Φ) are all counted by the same number Cat ( k ) (Φ). But more is true: definethe Fuß-Narayana number
N ar ( k ) (Φ , i ) as the number of elements of N C ( k ) (Φ)of rank n − i [Arm09, Definition 3.5.4.]. Let ( h , h , . . . , h n ) be the h -vector of∆ ( k ) (Φ), defined by the relation n X i =0 h i x n − i = X l,m f l,m ( x − n − ( l + m ) .Then h n − i = N ar ( k ) (Φ , i ) for all i ∈ { , , . . . , n } [FR05, Theorem 3.2.]. In thispaper, we prove the following conjecture of Fomin and Reading that relates theFuß-Narayana numbers to k -generalised nonnesting partitions as well. Theorem 1 ([FR05, Conjecture 10.1.]) . For a crystallographic root system Φ , thenumber of k -generalised nonnesting partitions of Φ that have exactly i indecompos-able elements of rank k equals the Fuß-Narayana number N ar ( k ) (Φ , i ) . The main result of this paper is the following theorem, conjectured by Armstrong.
Theorem 2 ([Arm09, Conjecture 5.3.2.]) . If Φ is a crystallographic root system ofrank n , then H k Φ ( x, y ) = ( x − n F k Φ (cid:18) x − , y − xx − (cid:19) . We first establish
Theorem 1 . Then we find a combinatorial bijection for k -generalised nonnesting partitions that leads to a differential equation for the H -triangle analogous to one known for the F -triangle. Using both of these differentialequations, Theorem 1 , and induction on the rank n , we prove Theorem 2 byshowing that the derivatives with respect to y of both sides of the equation as wellas their specialisations at y = 1 agree.After proving Theorem 2 , we use it to deduce various corollaries. Using the M = F (ex-)conjecture, we also relate the H -triangle to the M -triangle. Thisallows us to transfer a remarkable instance of combinatorial reciprocity observedby Krattenthaler [Kra06b, Theorem 8] for the M -triangle to the H -triangle. From Theorem 1 we also get a proof of a conjecture of Athanasiadis and Tzanaki [AT06,Conjecture 1.2.] that relates the h -vector of the positive part of ∆ ( k ) (Φ) to the enu-meration of bounded dominant regions in the k -Catalan arrangement of Φ by theirnumber of k -coloured ceilings.3. Proof of Theorem 1
Let N ( k ) (Φ , i ) be the number of k -generalised nonnesting partitions of Φ thathave exactly i indecomposable elements of rank k . If Φ is a classical root system, N ( k ) (Φ , i ) is known [Ath05, Proposition 5.1.] to equal the Fuß-Narayana number N ar ( k ) (Φ , i ). We wish to verify this also when Φ is of exceptional type. It sufficesto do this for Φ irreducible. Let n be the rank, h the Coxeter number and ˜ α thehighest root of Φ. Let L (Φ ∨ ) be the coroot lattice of Φ. Define A ◦ = { x ∈ V |h α i , x i ≥ α i ∈ S, h ˜ α, x i ≤ } , the fundamental simplex of Φ. In order toprove Theorem 1 , we use the following result, due to Athanasiadis.
Theorem 3 ([Ath05, Corollary 4.4.]) . The number of k -generalised nonnestingpartitions of Φ that have exactly i indecomposable elements of rank k equals thenumber of elements of the coroot lattice in the ( kh + 1) -fold dilation of the funda-mental simplex that are contained in exactly i of its walls. N THE H -TRIANGLE OF GENERALISED NONNESTING PARTITIONS 5 Proof of Theorem 1.
We wish to count elements of L (Φ ∨ ) ∩ tA ◦ by the number ofwalls they are contained in and then specialise to t = kh + 1. Now the simplecoroots { α ∨ , . . . , α ∨ n } are a Z -basis of L (Φ ∨ ). So if { e , . . . , e n } is the standardbasis of R n , the linear map Γ defined by Γ( α ∨ i ) = e i gives a lattice isomorphismfrom L (Φ ∨ ) to Z n . Let P = Γ( A ◦ ). Then P = { y ∈ Z n | (cid:10) α i , Γ − ( y ) (cid:11) ≥ α i ∈ S, (cid:10) ˜ α, Γ − ( y ) (cid:11) ≤ } = { y ∈ Z n | * α i , n X j =1 y j α ∨ j + ≥ α i ∈ S, * n X i =1 c i α i , n X j =1 y j α ∨ j + ≤ } = { y ∈ Z n | n X j =1 a ji y j ≥ α i ∈ S, n X i =1 c i n X j =1 a ji y j ≤ } = { y ∈ Z n | A T y ≥ α i ∈ S, c T A T y ≤ } .Here A = ( a ij ) = ( h α ∨ i , α j i ) is the Cartan matrix of Φ, and c = ( c , . . . , c n ) is thevector of coefficients in the simple root expansion of the highest root ˜ α = P ni =1 c i α i .Now since Γ is an isomorphism, it maps walls to walls, so instead of counting el-ements of L (Φ ∨ ) ∩ tA ◦ by the number of walls they are contained in we countelements of Γ( L (Φ ∨ ) ∩ tA ◦ ) = Z n ∩ tP by the number of walls they are containedin.Let B be a face of P , and let f B ( t ) = | Z n ∩ tB | be the number of lattice points in tB . Let g B ( t ) be the number of lattice points in tB that are in no other face of P contained in B . Then by the Inclusion-Exclusion Principle, g B ( t ) = X C ⊆ B ( − dim ( B ) − dim ( C ) f C ( t ).Using the Macaulay2 interface of Normaliz [BK10], for any irreducible exceptionalroot system Φ, we calculate the Ehrhart series F B ( z ) = P ∞ t =0 f B ( t ) z t for all faces B of P . This is a rational function in z , since the vertices of P are in Q n . Fromthis we get G B ( z ) = ∞ X t =0 g B ( t ) z t = X C ⊆ B ( − dim ( B ) − dim ( C ) F C ( z ).Then we define N i ( z ) = P B G B ( z ), where the sum is over all faces B of P ofdimension n − i . So the number of lattice points in tP that are contained in exactly i walls of tP is equal to [ z t ] N i ( z ), the coefficient of z t in the power series N i ( z ). So N ( k ) (Φ , i ) = [ z kh +1 ] N i ( z ). Recall from Ehrhart theory that [ z t ] F B ( z ), and thereforealso [ z t ] G B ( z ) and [ z t ] N i ( z ), are quasipolynomials in t , with degree at most n andperiod p , the least common multiple of the denominators of the vertices of P . Thus N ( k ) (Φ , i ) = [ z kh +1 ] N i ( z ) is a quasipolynomial in k , with period p ′ = lcm ( p,h ) h . Itturns out that p ′ is either 1 or 2 in every case. Since N ar ( k ) (Φ , i ) is a polynomial in k of degree at most n , in order to verify that N ( k ) (Φ , i ) = N ar ( k ) (Φ , i ) for all i and k , one only needs to check for every i that they agree for the first ( n + 1) p ′ values.Since the generating function N i ( z ) has been computed and explicit formulae for N ar ( k ) (Φ , i ) are known [Arm09, Theorem 3.5.6.], this is easily accomplished. (cid:3) The Bijection
The other main tool in the proof of
Theorem 2 is the following bijection.
MARKO THIEL
Theorem 4.
For every simple root α ∈ S , there exists a bijection Θ from the set of k -generalised nonnesting partitions I ∈ N N ( k ) (Φ( S )) with α ∈ I k to N N ( k ) (Φ( S \{ α } )) .The rank l indecomposable elements of Θ( I ) are exactly the rank l indecomposableelements of I if l < k . The rank k indecomposable elements of Θ( I ) are exactly therank k indecomposable elements of I except for α . In order to prove this, we first need a basic lemma, implicit in [Ath05].
Lemma 1.
The rank l indecomposable elements of a k -generalised nonnesting par-titions I ∈ N N ( k ) (Φ) are minimal elements of I l .Proof. Let α ∈ I l be an indecomposable element. Suppose for contradiction that α is not minimal in I l , say α > β ∈ I l . Then α = β + P mi =1 α i , where α i ∈ S for all i ∈ [ m ]. So P mi =1 α i ∈ Φ or β + P i = j α i ∈ Φ for some j ∈ [ m ], by [Ath05, Lemma2.1. (i)]. In the first case, α = β + P mi =1 α i , with β ∈ I l and P mi =1 α i ∈ V , so α isnot indecomposable. In the second case, α = β + P i = j α i + α j , with β + P i = j α i ∈ I l and α j ∈ I , so α is not indecomposable. (cid:3) Proof of Theorem 4.
Let I ( α ) be the order filter in the root poset generated by α ,that is the set of all positive roots that have α in their support. Define θ ( I i ) = I i \ I ( α ) = I i ∩ Φ( S \{ α } ), where Φ( S \{ α } ) is the root system with simple system S \{ α } . Then let Θ( I ) = ( θ ( I ) , θ ( I ) , . . . , θ ( I k )).We claim that Θ( I ) is a k -generalised nonnesting partition of Φ( S \{ α } ) and thusΘ is well-defined.In order to see this, first observe that every θ ( I i ) in Θ( I ) is an order filter inthe root poset of Φ( S \{ α } ), and the θ ( I i ) form a (multi)chain under inclusion. Forall i, j ∈ { , , . . . , k } ,( θ ( I i ) + θ ( I j )) ∩ Φ + ( S \{ α } ) ⊆ ( I i + I j ) ∩ Φ + ( S \{ α } ) ⊆ I i + j ∩ Φ + ( S \{ α } ) = θ ( I i + j ).Also θ ( J i ) = J i for all i ∈ { , , . . . , k } , so ( θ ( J i ) + θ ( J j )) ∩ Φ + ( S \{ α } ) ⊆ θ ( J i + j )for all i, j with i + j ≤ k . So Θ( I ) is a geometric (multi)chain of order fiters in theroot order of Φ( S \{ α } ), and the claim follows.Now define a map Ψ from N N ( k ) (Φ( S \{ α } )) to the set of I ∈ N N ( k ) (Φ( S )) with α ∈ I k by ψ ( I i ) = I i ∪ I ( α ), and Ψ( I ) = ( ψ ( I ) , ψ ( I ) , . . . , ψ ( I k )).We claim that Ψ( I ) is a k -generalised nonnesting partition of Φ( S ) and thus Ψis well-defined.In order to see this, first observe that every ψ ( I i ) is an order filter in the root poset ofΦ( S ) and the ψ ( I i ) form a (multi)chain under inclusion. For all i, j ∈ { , , . . . , k } ,( ψ ( I i ) + ψ ( I j )) ∩ Φ + ( S ) = (( I i ∪ I ( α )) + ( I j ∪ I ( α )) ∩ Φ + ( S ) ⊆ (( I i + I j ) ∩ Φ + ( S )) ∪ I ( α ) ⊆ I i + j ∪ I ( α ) = ψ ( I i + j ).Also ψ ( J i ) = J i for all i ∈ { , , . . . , k } , so( ψ ( J i ) + ψ ( J j )) ∩ Φ + ( S ) = ( ψ ( J i ) + ψ ( J j )) ∩ Φ + ( S \{ α } ) ⊆ ψ ( J i + j )for all i, j with i + j ≤ k . So Θ( I ) is a geometric (multi)chain of order fiters in theroot order of Φ( S ), and the claim follows.Now Θ and Ψ are inverse to each other, so Θ is a bijection, as required. N THE H -TRIANGLE OF GENERALISED NONNESTING PARTITIONS 7 We claim that for β ∈ Φ + , β is a rank l indecomposable element in Θ( I ) if andonly if β is a rank l indecomposable element in I and β = α .In order to see this, first notice that for β ∈ θ ( I l ), k β (Θ( I )) = k β ( I ). The onlyelement in I l \ θ ( I l ) = I ( α ) that can be indecomposable of rank l is α , since all otherelements are not minimal, so not indecomposable by Lemma 1 . So if β = α is arank l indecomposable element of V , then β ∈ θ ( I l ). If β were not indecomposablein Θ( I ), then either β = γ + δ for γ ∈ θ ( I i ), δ ∈ θ ( I j ), with i + j = l , in contradictionto β being indecomposable in I , or there is a γ / ∈ θ ( I t − l ) with β + γ ∈ θ ( I t ) and k β + γ (Θ( I )) = t , for some l ≤ t ≤ k , also in contradiction to β being indecompos-able in I . So β is rank l indecomposable in Θ( I ).Now for β a rank l indecomposable element of Θ( I ), suppose for contradictionthat β were not indecomposable in I . If β = γ + δ for γ ∈ I i , δ ∈ I j , with i + j = l , then α is not in the support of either γ or δ , so γ ∈ θ ( I i ) and δ ∈ θ ( I j ),a contradiction to β being indecomposable in Θ( I ). If β + γ ∈ I t and k β + γ ( I ) = t for some l ≤ t ≤ k , and γ ∈ Φ( S \{ α } ), then γ ∈ θ ( I t ) ⊆ I t , as β is indecomposablein Θ( I ). If β + γ ∈ I t for some l ≤ t ≤ k and γ / ∈ Φ( S \{ α } ), then γ ∈ I ( α ), so γ ∈ I k ⊆ I t − l . So β is indecomposable in I . This establishes the claim.Thus Θ is a bijection having the desired properties. (cid:3) Proof of the Main Result
To prove
Theorem 2 , we show that the derivatives with respect to y of bothsides of the equation agree, as well as their specialisations at y = 1. To do this, weneed the following lemmas. Lemma 2. If Φ is a crystallographic root system of rank n , then H k Φ ( x,
1) = ( x − n F k Φ (cid:18) x − , x − (cid:19) .Proof. We have( x − n F k Φ (cid:18) x − , x − (cid:19) = X l,m f l,m ( x − n − ( l + m ) = n X i =0 h i x n − i ,where ( h , h , . . . , h n ) is the h -vector of ∆ ( k ) (Φ). So[ x i ]( x − n F k Φ (cid:18) x − , x − (cid:19) = h n − i = N ar ( k ) (Φ , i ),by [FR05, Theorem 3.2.]. But[ x i ] H k Φ ( x,
1) =
N ar ( k ) (Φ , i ),by Theorem 1 . (cid:3) Lemma 3 ([Kra06a, Proposition F (2)]) . If Φ is a crystallographic root system ofrank n , then ∂∂y F k Φ( S ) ( x, y ) = X α ∈ S F k Φ( S \{ α } ) ( x, y ) . MARKO THIEL
Proof.
As mentioned in [Kra06a], this can be proven in the same way as the k = 1case, where it is due to Chapoton [Cha04, Proposition 3]. For completeness, as wellas to highlight the analogy to the proof of Lemma 4 , we give the proof here.We wish to show that mf kl,m (Φ) = X α ∈ S f kl,m − (Φ( S \{ α } )),that is, we seek a bijection ϕ from the set of pairs ( A, − α ) with A ∈ ∆ ( k ) (Φ( S ))and − α ∈ A ∩ ( − S ) to ∐ α ∈ S ∆ ( k ) (Φ( S \{ α } )) such that ϕ ( A ) contains the samenumber of coloured positive roots as A , but exactly one less uncoloured negativesimple root. By [FR05, Proposition 3.5.], the map ϕ ( A, − α ) = ( A \{− α } , α ) is sucha bijection. (cid:3) Lemma 4. If Φ is a crystallographic root system of rank n , then ∂∂y H k Φ( S ) ( x, y ) = x X α ∈ S H k Φ( S \{ α } ) ( x, y ) .Proof. Analogously to
Lemma 3 , we seek a bijection Θ from the set of pairs (
I, α )with I ∈ N N ( k ) (Φ( S )) and α ∈ I k ∩ S to ∐ α ∈ S N N ( k ) (Φ( S \{ α } )) such that Θ( I ) hasexactly one less rank k simple element and exactly one less rank k indecomposableelement than I . Such a bijection is given in Theorem 4 . (cid:3) We are now in a position to prove
Theorem 2 . Proof of Theorem 2.
We proceed by induction on n . If n = 0, both sides are equalto 1, so the result holds. If n > ∂∂y H k Φ( S ) ( x, y ) = x X α ∈ S H k Φ( S \{ α } ) ( x, y ),by Lemma 4 . By induction hypothesis, this is further equal to x X α ∈ S ( x − n − F k Φ( S \{ α } ) (cid:18) x − , y − xx − (cid:19) ,which equals ∂∂y ( x − n F k Φ( S ) (cid:18) x − , y − xx − (cid:19) by Lemma 3 . But H k Φ ( x,
1) = ( x − n F k Φ (cid:18) x − , x − (cid:19) by Lemma 2 , so H k Φ ( x, y ) = ( x − n F k Φ (cid:18) x − , y − xx − (cid:19) ,since the derivatives with respect to y as well as the specialisations at y = 1 of bothsides agree. (cid:3) N THE H -TRIANGLE OF GENERALISED NONNESTING PARTITIONS 9 Corollaries of the Main Result
Specialising
Theorem 2 to k = 1, we can now prove Chapoton’s original con-jecture. Corollary 1 ([Cha06, Conjecture 6.1.]) . If Φ is a crystallographic root system ofrank n , then H ( x, y ) = (1 − x ) n F (cid:18) x − x , xy − x (cid:19) .Proof. We have H ( x, y ) = ( x − n F (cid:18) x − , y − xx − (cid:19) .(1)But we also have [Cha04, Proposition 5] F ( x, y ) = ( − n F ( − − x, − − y ).(2)Substituting (2) into (1), we obtain H ( x, y ) = (1 − x ) n F (cid:18) x − x , xy − x (cid:19) . (cid:3) Using the M = F (ex-)conjecture, we can also relate the H -triangle to the M -triangle. Corollary 2 ([Arm09, Conjecture 5.3.2.]) . If Φ is a crystallographic root systemof rank n , then H k Φ ( x, y ) = (1 + ( y − x ) n M k Φ (cid:18) yy − , ( y − x y − x (cid:19) .Proof. We have H k Φ ( x, y ) = ( x − n F k Φ (cid:18) x − , y − xx − (cid:19) .(3)But we also have [Kra06a, Conjecture FM] [Tza08, Theorem 1.2.] F k Φ ( x, y ) = y n M k Φ (cid:18) yy − x , y − xy (cid:19) .(4)Substituting (4) into (3), we obtain H k Φ ( x, y ) = (1 + ( y − x ) n M k Φ (cid:18) yy − , ( y − x y − x (cid:19) . (cid:3) The coefficients of F k Φ ( x, y ) are known to be polynomials in k [Kra06a], so thecoefficients of H k Φ ( x, y ) are also polynomials in k . Thus it makes sense to consider H k Φ ( x, y ) even if k is not a positive integer. We can use Corollary 2 to trans-fer a remarkable instance of combinatorial reciprocity observed by Krattenthaler[Kra06b, Theorem 8] for the M -triangle to the H -triangle. Corollary 3. If Φ is a crystallographic root system of rank n , then H k Φ ( x, y ) = ( − n H − k Φ (cid:18) − x, − xy − x (cid:19) . Proof.
We have H k Φ ( x, y ) = (1 + ( y − x ) n M k Φ (cid:18) yy − , ( y − x y − x (cid:19) .(5)But we also have [Kra06b, Theorem 8] [Tza08, Theorem 1.2.] M k Φ ( x, y ) = y n M − k Φ (cid:18) xy, y (cid:19) .(6)Substituting (6) into (5), we obtain H k Φ ( x, y ) = (( y − x ) n M − k Φ (cid:18) xy y − x , y − x ( y − x (cid:19) .(7)Inverting (5), we get M k Φ ( x, y ) = (1 − y ) n H k Φ (cid:18) y ( x − − y , xx − (cid:19) .(8)Substituting (8) into (7), we obtain H k Φ ( x, y ) = ( − n H − k Φ (cid:18) − x, − xy − x (cid:19) . (cid:3) For k = 1, we can transfer a duality for the F -triangle to the H -triangle. Corollary 4. H ( x, y ) = x n H (cid:18) x , y − x (cid:19) .Proof. Inverting
Theorem 2 , we get F ( x, y ) = x n H (cid:18) x + 1 x , y + 1 x + 1 (cid:19) .(9)Thus H ( x, y ) = (1 − x ) n F (cid:18) x − x , xy − x (cid:19) = x n H (cid:18) x , y − x (cid:19) ,using Corollary 1 and (9). (cid:3)
For a bounded dominant region R in the k -Catalan arrangement, let CL k ( R )be the number of k -coloured ceilings of R . Let h + i (Φ) be the number of boundeddominant regions R that have exactly n − i k -coloured ceilings. Let h i (∆ k + (Φ)) bethe i -th entry of the h -vector of the positive part of ∆ ( k ) (Φ). Corollary 5 ([AT06, Conjecture 1.2.]) . If Φ is a crystallographic root system, h + i (Φ) = h i (∆ k + (Φ)) for all i .Proof. This follows from
Theorem 1 and [AT06, Corollary 5.4.]. (cid:3)
Corollary 6. If Φ is a crystallographic root system, then H k Φ (cid:18) x, − x (cid:19) = X R x | CL k ( R ) | ,where the sum is over all bounded dominant regions R in the k -Catalan arrangementof Φ . N THE H -TRIANGLE OF GENERALISED NONNESTING PARTITIONS 11 Proof. X R x | CL k ( R ) | = n X i =0 h + i (Φ) x n − i = n X i =0 h i (∆ k + (Φ)) x n − i = n X l =0 f kl, (Φ)( x − n − l = ( x − n F k Φ (cid:18) x − , (cid:19) = H k Φ (cid:18) x, − x (cid:19) ,using Corollary 5 and
Theorem 2 . (cid:3) Corollary 7. If Φ is a crystallographic root system of rank n , then X R x | CL ( R ) | = x n H (cid:18) x , (cid:19) ,where the sum is over all bounded dominant regions R in the -Catalan arrangementof Φ .Proof. X R x | CL ( R ) | = H (cid:18) x, − x (cid:19) = x n H (cid:18) x , (cid:19) ,using Corollary 6 and
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