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ν-Tamari lattices via subword complexes
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Electronic Notes in Discrete Mathematics 61 (2017) 215–221 www.elsevier.com/locate/endm
ν-Tamari lattices via subword complexes Cesar Ceballos 1,3 Faculty of Mathematics, University of Vienna, Vienna, Austria
Arnau Padrol 2 Sorbonne Universit´es, Universit´e Pierre et Marie Curie (Paris 6), Institut de Math´ematiques de Jussieu - Paris Rive Gauche (UMR 7586), Paris, France
Camilo Sarmiento Universit¨ at Leipzig, Mathematisches Institut, Leipzig, Germany
Abstract We show that the ν-Tamari lattices of Pr´eville-Ratelle and Viennot can obtained as the duals of certain subword complexes. This generalizes a known result for the classical Tamari lattice, provides a simple description of the lattice property using certain bracket vectors of ν-trees, and gives (conjectural) insight on the geometry of more general objects in terms of polytopal subdivisions of multiassociahedra. Keywords: Tamari lattices, Subword complexes, Rooted binary trees
1
Supported by the Austrian Science Foundation FWF, grant F 5008-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. 2 Supported by the program PEPS Jeunes Chercheur-e-s from the INSMI. 3 Email: [email protected] http://dx.doi.org/10.1016/j.endm.2017.06.041 1571-0653/© 2017 Elsevier B.V. All rights reserved.
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C. Ceballos et al. / Electronic Notes in Discrete Mathematics 61 (2017) 215–221
Introduction
The Tamari lattice is a partial order on Catalan objects that has been widely studied since it was first introduced by Tamari in his doctoral thesis in 1951 [13]. It has been generalized to m-Tamari lattices by F. Bergeron and Pr´evilleRatelle [1], and further generalized to ν-Tamari lattices by Pr´eville-Ratelle and Viennot [10]. These lattices have attracted considerable attention in other areas such as representation theory and Hopf algebras [3,7], and their origin is motivated by their conjectural connections with dimension formulas in trivariate diagonal harmonics [1]. In [9], [8, Theorem 3.18] Pilaud and Pocchiola discovered a striking relation between the classical Tamari lattice and Knutson and Miller’s theory of subword complexes [6]. A particular case of their result was rediscovered by Stump [12] and Stump and Serrano in [11], and it also follows from Woo’s earlier bijection between certain pipe dreams and Dyck paths in [14, Section 3]. Theorem 1.1 ([9,11,12,14]) The Hasse diagram of the classical Tamari lattice is isomorphic to the facet adjacency graph of certain subword complex. Subword complexes are simplicial complexes described in Definition 2.1. The facet adjacency graph of a complex has its facets as vertices and adjacent facets as edges. Two facets are adjacent if they share a codimension-1 face. Theorem 1.1 has inspired further connections with other topics such as multi-associahedra, pseudotriangulation polytopes, cluster algebras and Hopf algebras, see for example [2,4] and the references therein. In this paper we extend Theorem 1.1 in the context of ν-Tamari lattices. Our realization of the ν-Tamari lattice as the dual of a subword complex (Theorem 2.2) uses a notion of ν-trees and a natural rotation operation that endows them with a lattice structure (Theorem 3.2). This approach provides a simple description of the lattice property using certain bracket vectors of ν-trees (Theorem 4.2), and gives (conjectural) insight on the geometry of more general objects in terms of polytopal subdivisions of multiassociahedra (Section 5).
2
ν-Tamari lattices and subword complexes
We identify lattice paths that start at (0, 0) and consist of a finite number of north and east unit steps with words on the alphabet {N, E}. Given a lattice path ν, a ν-path is a lattice path with the same endpoints as ν that is weakly above ν. The following lattice structure on ν-paths was introduced in [10]. Let μ be a ν-Dyck path. For a lattice point p on μ let horizν (p) be the
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maximum number of horizontal steps that can be added to the right of p without crossing ν. Given a valley p of μ (a point preceded by an east step E and followed by a north step N) let q be the first lattice point in μ after p such that horizν (q) = horizν (p). We denote by μ[p,q] the subpath of μ that starts at p and finishes at q, and consider the path μ obtained from μ by switching E and μ[p,q] . The covering relation is defined by μ <ν μ . The ν-Tamari lattice Tamν is the transitive closure of this relation. An example is illustrated in Figure 1 (left). The case ν = (NE)n yields the classical Tamari lattice. 2
2
2
2 1
(2,0,2,2,1,2)
0
2
2
2
2 2
0
1 0
1
1
1
1
(2,0,1,1,1,2)
(0,0,2,1,1,2)
2
0
(2,0,2,1,1,2)
(0,0,2,2,1,2)
0 1
1
2 1
2
2
2
0
1 0
2
1
1
(1,0,1,1,1,2)
1
0 2
(0,0,1,1,1,2)
1 0
1
1
0
Fig. 1. The ν-Tamari lattice for ν = ENEEN (left). The lattice of ν-trees under rotation (middle). The lattice of ν-bracket vectors (right).
Let Sn+1 be the symmetric group of permutations of [n + 1], and S = {s1 , . . . , sn } be the generating set of simple transpositions si = (i i+1). Every element w ∈ Sn+1 can be written as a product w = si1 si2 . . . sik of elements in S. If k is minimal among all such expressions for w, then k is called the length (w) of w, and si1 si2 . . . sik is called a reduced expression for w. Definition 2.1 [[6]] Let Q = (q1 , . . . , qm ) be a word in S and π ∈ Sn+1 be some permutation. The subword complex SC(Q, π) is the simplicial complex whose facets are given by the subsets I ⊂ [m] = {1, 2, . . . , m}, such that the subword of Q with positions at [m] I is a reduced expression of π. 4 Theorem 2.2 The Hasse diagram of the ν-Tamari lattice is isomorphic to the facet adjacency graph of certain subword complex SC(Qν , πν ). The permutation πν and the word Qν can be described as follows. Let λν be the partition bounded by ν, and for every lattice point p in λν denote 4
Knutson and Miller’s definition of subword complexes work for arbitrary Coxeter groups, but in this paper we only need to consider the case of the symmetric group.
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4
3
5
2
6
s1 s2 s3 s4 s2 s3 s4 s5 s3 s4
Fig. 2. Lattice path ν = ENEEN and the partition λν it bounds (left). Rothe diagram of the permutation πν = [1, 4, 3, 5, 2, 6] (middle). Corresponding word Qν = (s3 , s2 , s1 , s4 , s3 , s2 , s4 , s3 , s5 , s4 ) (right).
by d(p) the lattice distance from p to the top-left corner of λν . Set dˆ = maxp∈λν d(p). Then πν is the permutation in Sd+2 whose Rothe diagram (i.e. ˆ the set {(π(j), i) : i < j, π(i) > π(j)}) is the partition λν with its northwest block lying at (2,2). Now label each integer lattice point p in λν by the transposition sd(p)+1 . The word Qν is obtained by reading the labels from bottom to top, from left to right. See Figure 2 (compare [11]).
3
The rotation lattice of ν-trees
The principal ingredient towards our main result is the description of Tamν in terms of ν-trees, which are in bijection with binary trees of canopy ν (cf. [10]). Let V be the set of lattice points inside the partition λν bounded by ν. We say that p, q ∈ V are not ν-compatible if and only if p and q form a strict north-east chain (i.e. either q − p or p − q has strictly positive entries) that is contained in a rectangle that fits weakly above ν. A ν-tree is a maximal collection of pairwise ν-compatible elements in V . Two ν-trees T, T are related by a rotation if T can be obtained by exchanging an element q ∈ T by an element q ∈ T as illustrated in Figure 3, where p, r are both in T and T , and there are no further points of T or T on the dotted lines. The rotation is a right rotation if q is north-east from q, and a left rotation otherwise. p p q q
r
r
Fig. 3. Rotation operation on ν-trees.
Let T be a ν-tree. By connecting each element p of T with the first element
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of T below it (if any), and with the first element of T to its right (if any), we can view T as a rooted binary tree in the graph-theoretical sense; see Figure 1 (middle). This also makes our notion of ν-tree rotation more self-explanatory. Lemma 3.1 A rotation of a ν-tree is also a ν-tree. Strikingly, the complements of ν-trees determine all reduced expressions of πν in Qν [11, Theorem 2.1]. Moreover, two facets are adjacent if and only if the corresponding ν-trees are related by a rotation. Figure 4 illustrates an example. Theorem 2.2 then follows from the following main result. Theorem 3.2 The ν-Tamari lattice is isomorphic to the rotation lattice of ν-trees, whose covering relation is given by right rotation. s2 s2
s4
s4
rotation
s3
s2 s3 s2 s4 = [1, 4, 3, 5, 2, 6]
s2
s3
s3 s3 s2 s3 s4 = [1, 4, 3, 5, 2, 6]
Fig. 4. Complements of ν-trees are the reduced expressions of πν in Qν .
4
A simple proof of the lattice property
The meet and join operations in the ν-Tamari lattice turn out to be very simple when working with ν-trees instead of ν-paths. Definition 4.1 Let ν be a lattice path from (0, 0) to (a, b) with length = (ν) = a + b. The minimal ν-bracket vector bmin is a vector consisting of +1 non-negative integers obtained by reading the y-coordinates of the lattice points on ν in the order they appear along the path. We define the set of fixed positions as the set F = {f0 , f1 , . . . , fb } where fk is the position of the last appearance of k in bmin . A ν-bracket vector is a vector b = (b1 , . . . , b+1 ) satisfying the following properties: (i) bfk = k for 0 ≤ k ≤ b; (ii) bmin ≤ bi ≤ b for all i; i (iii) if bi = k, then bj ≤ k for i ≤ j ≤ fk . Theorem 4.2 The poset of ν-bracket vectors under componentwise order is a lattice isomorphic to the ν-Tamari lattice. The ν-bracket vectors can be obtained from traversals of ν-trees. Recall that the in-order traversal of a binary tree is defined recursively as follows: if
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x is a node with is left subtree A and right subtree B, we traverse the nodes of A in in-order, then x and finally traverse B in in-order. Definition 4.3 We label each node of a ν-tree T by the entry of its ycoordinate. The bracket vector b(T ) is the result of reading the labels of the nodes in in-order. See Figure 1. The meet operation can be easily described in terms of ν-bracket vectors: Proposition 4.4 The meet on the set of ν-bracket vectors is given by componentwise minimum. That is b ∧ b = (min{b1 , b1 }, . . . , min{b+1 , b+1 }).
5
Multi ν-Tamari complexes
For any integer k ≥ 1 one may define the (k, ν)-Tamari complex as the simplicial complex on V whose minimal non-faces are (k + 1)-subsets of pairwise non ν-compatible elements. For ν = (N E)n this coincides with the simplicial complex of (k + 1)-crossing-free subsets of diagonals of a convex (n + 2)-gon, whose facet adjacency graph is conjectured to be realizable as the edge graph of a simple polytope: the multiassociahedron Δn+2,k . For k = 1 we have recently shown that the facet adjacency graph of (1, ν)-Tamari complexes can be realized as the edge graphs of polytopal subdivisions of associahedra 5 [5]. We believe that a similar statement might be true for general k. The following proposition is a first positive result in this direction:
Fig. 5. Facet adjacency graph of the (k, ν)-Tamari complex for k = 2 and ν = (NE5 )3 (left), and k = 3 and ν = (NE5 )4 (right).
Proposition 5.1 Let m ≥ k and ν = (N E m )k+1 . The facet adjacency graph Gk,ν of the Fuss-Catalan (k, ν)-Tamari complex is the edge graph of a polytopal subdivision of the multi-associahedron Δ2k+2,k . More precisely, Δ2k+2,k is a k-dimensional simplex, and Gk,ν is the edge graph of the staircase subdivision of its (m − k + 1) dilation, see Figure 5. 5
Whenever ν does not have two consecutive non-initial north steps
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References [1] Bergeron, F. and L.-F. Pr´eville-Ratelle, Higher trivariate diagonal harmonics via generalized Tamari posets, J. Comb. 3 (2012), pp. 317–341. [2] Bergeron, N. and C. Ceballos, A Hopf algebra of subword complexes, Advances in Mathematics 305 (2017), pp. 1163–1201. [3] Bousquet-M´elou, M., G. Chapuy and L.-F. Pr´eville-Ratelle, The representation of the symmetric group on m-Tamari intervals, Adv. Math. 247 (2013), pp. 309–342. [4] Ceballos, C., J.-P. Labb´e and C. Stump, Subword complexes, cluster complexes, and generalized multi-associahedra, J. Algebraic Combin., 39(1):17–51, 2014. [5] Ceballos, C., A. Padrol and C. Sarmiento, Geometry of ν-Tamari lattices in types a and b, arXiv preprint arXiv:1611.09794 (2016). [6] Knutson, A. and E. Miller, Subword complexes in Coxeter groups, Adv. Math. 184 (2004), pp. 161–176. [7] Novelli, J.-C. and J.-Y. Thibon, Hopf algebras of m-permutations,(m+ 1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 (2014). [8] Pilaud, V., “Multitriangulations, pseudotriangulations et quelques problemes de r´ealisation de polytopes,” Ph.D. thesis, Paris 7 (2010). [9] Pilaud, V. and M. Pocchiola, Multitriangulations, pseudotriangulations and primitive sorting networks, Discrete Comput. Geom. 48 (2012), pp. 142–191. [10] Pr´eville-Ratelle, L.-F. and X. Viennot, An extension of Tamari lattices, to appear in Trans. Amer. Math. Soc. [11] Serrano, L. and C. Stump, Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials, Electron. J. Combin. 19 (2012), p. P16. [12] Stump, C., A new perspective on k-triangulations, J. Combin. Theory Ser. A 118 (2011), pp. 1794–1800. [13] Tamari, D., “Mono¨ıdes pr´eordonn´es et chaˆınes de Malcev,” Ph.D. thesis, Sorbonne Paris (1951). [14] Woo, A., Catalan numbers and Schubert polynomials for w = 1(n + 1)...2, preprint, Jul. 2004, arXiv:0407160.
Electronic Notes in Discrete Mathematics 61 (2017) 215–221 www.elsevier.com/locate/endm
ν-Tamari lattices via subword complexes Cesar Ceballos 1,3 Faculty of Mathematics, University of Vienna, Vienna, Austria
Arnau Padrol 2 Sorbonne Universit´es, Universit´e Pierre et Marie Curie (Paris 6), Institut de Math´ematiques de Jussieu - Paris Rive Gauche (UMR 7586), Paris, France
Camilo Sarmiento Universit¨ at Leipzig, Mathematisches Institut, Leipzig, Germany
Abstract We show that the ν-Tamari lattices of Pr´eville-Ratelle and Viennot can obtained as the duals of certain subword complexes. This generalizes a known result for the classical Tamari lattice, provides a simple description of the lattice property using certain bracket vectors of ν-trees, and gives (conjectural) insight on the geometry of more general objects in terms of polytopal subdivisions of multiassociahedra. Keywords: Tamari lattices, Subword complexes, Rooted binary trees
1
Supported by the Austrian Science Foundation FWF, grant F 5008-N15, in the framework of the Special Research Program “Algorithmic and Enumerative Combinatorics”. 2 Supported by the program PEPS Jeunes Chercheur-e-s from the INSMI. 3 Email: [email protected] http://dx.doi.org/10.1016/j.endm.2017.06.041 1571-0653/© 2017 Elsevier B.V. All rights reserved.
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C. Ceballos et al. / Electronic Notes in Discrete Mathematics 61 (2017) 215–221
Introduction
The Tamari lattice is a partial order on Catalan objects that has been widely studied since it was first introduced by Tamari in his doctoral thesis in 1951 [13]. It has been generalized to m-Tamari lattices by F. Bergeron and Pr´evilleRatelle [1], and further generalized to ν-Tamari lattices by Pr´eville-Ratelle and Viennot [10]. These lattices have attracted considerable attention in other areas such as representation theory and Hopf algebras [3,7], and their origin is motivated by their conjectural connections with dimension formulas in trivariate diagonal harmonics [1]. In [9], [8, Theorem 3.18] Pilaud and Pocchiola discovered a striking relation between the classical Tamari lattice and Knutson and Miller’s theory of subword complexes [6]. A particular case of their result was rediscovered by Stump [12] and Stump and Serrano in [11], and it also follows from Woo’s earlier bijection between certain pipe dreams and Dyck paths in [14, Section 3]. Theorem 1.1 ([9,11,12,14]) The Hasse diagram of the classical Tamari lattice is isomorphic to the facet adjacency graph of certain subword complex. Subword complexes are simplicial complexes described in Definition 2.1. The facet adjacency graph of a complex has its facets as vertices and adjacent facets as edges. Two facets are adjacent if they share a codimension-1 face. Theorem 1.1 has inspired further connections with other topics such as multi-associahedra, pseudotriangulation polytopes, cluster algebras and Hopf algebras, see for example [2,4] and the references therein. In this paper we extend Theorem 1.1 in the context of ν-Tamari lattices. Our realization of the ν-Tamari lattice as the dual of a subword complex (Theorem 2.2) uses a notion of ν-trees and a natural rotation operation that endows them with a lattice structure (Theorem 3.2). This approach provides a simple description of the lattice property using certain bracket vectors of ν-trees (Theorem 4.2), and gives (conjectural) insight on the geometry of more general objects in terms of polytopal subdivisions of multiassociahedra (Section 5).
2
ν-Tamari lattices and subword complexes
We identify lattice paths that start at (0, 0) and consist of a finite number of north and east unit steps with words on the alphabet {N, E}. Given a lattice path ν, a ν-path is a lattice path with the same endpoints as ν that is weakly above ν. The following lattice structure on ν-paths was introduced in [10]. Let μ be a ν-Dyck path. For a lattice point p on μ let horizν (p) be the
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maximum number of horizontal steps that can be added to the right of p without crossing ν. Given a valley p of μ (a point preceded by an east step E and followed by a north step N) let q be the first lattice point in μ after p such that horizν (q) = horizν (p). We denote by μ[p,q] the subpath of μ that starts at p and finishes at q, and consider the path μ obtained from μ by switching E and μ[p,q] . The covering relation is defined by μ <ν μ . The ν-Tamari lattice Tamν is the transitive closure of this relation. An example is illustrated in Figure 1 (left). The case ν = (NE)n yields the classical Tamari lattice. 2
2
2
2 1
(2,0,2,2,1,2)
0
2
2
2
2 2
0
1 0
1
1
1
1
(2,0,1,1,1,2)
(0,0,2,1,1,2)
2
0
(2,0,2,1,1,2)
(0,0,2,2,1,2)
0 1
1
2 1
2
2
2
0
1 0
2
1
1
(1,0,1,1,1,2)
1
0 2
(0,0,1,1,1,2)
1 0
1
1
0
Fig. 1. The ν-Tamari lattice for ν = ENEEN (left). The lattice of ν-trees under rotation (middle). The lattice of ν-bracket vectors (right).
Let Sn+1 be the symmetric group of permutations of [n + 1], and S = {s1 , . . . , sn } be the generating set of simple transpositions si = (i i+1). Every element w ∈ Sn+1 can be written as a product w = si1 si2 . . . sik of elements in S. If k is minimal among all such expressions for w, then k is called the length (w) of w, and si1 si2 . . . sik is called a reduced expression for w. Definition 2.1 [[6]] Let Q = (q1 , . . . , qm ) be a word in S and π ∈ Sn+1 be some permutation. The subword complex SC(Q, π) is the simplicial complex whose facets are given by the subsets I ⊂ [m] = {1, 2, . . . , m}, such that the subword of Q with positions at [m] I is a reduced expression of π. 4 Theorem 2.2 The Hasse diagram of the ν-Tamari lattice is isomorphic to the facet adjacency graph of certain subword complex SC(Qν , πν ). The permutation πν and the word Qν can be described as follows. Let λν be the partition bounded by ν, and for every lattice point p in λν denote 4
Knutson and Miller’s definition of subword complexes work for arbitrary Coxeter groups, but in this paper we only need to consider the case of the symmetric group.
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4
3
5
2
6
s1 s2 s3 s4 s2 s3 s4 s5 s3 s4
Fig. 2. Lattice path ν = ENEEN and the partition λν it bounds (left). Rothe diagram of the permutation πν = [1, 4, 3, 5, 2, 6] (middle). Corresponding word Qν = (s3 , s2 , s1 , s4 , s3 , s2 , s4 , s3 , s5 , s4 ) (right).
by d(p) the lattice distance from p to the top-left corner of λν . Set dˆ = maxp∈λν d(p). Then πν is the permutation in Sd+2 whose Rothe diagram (i.e. ˆ the set {(π(j), i) : i < j, π(i) > π(j)}) is the partition λν with its northwest block lying at (2,2). Now label each integer lattice point p in λν by the transposition sd(p)+1 . The word Qν is obtained by reading the labels from bottom to top, from left to right. See Figure 2 (compare [11]).
3
The rotation lattice of ν-trees
The principal ingredient towards our main result is the description of Tamν in terms of ν-trees, which are in bijection with binary trees of canopy ν (cf. [10]). Let V be the set of lattice points inside the partition λν bounded by ν. We say that p, q ∈ V are not ν-compatible if and only if p and q form a strict north-east chain (i.e. either q − p or p − q has strictly positive entries) that is contained in a rectangle that fits weakly above ν. A ν-tree is a maximal collection of pairwise ν-compatible elements in V . Two ν-trees T, T are related by a rotation if T can be obtained by exchanging an element q ∈ T by an element q ∈ T as illustrated in Figure 3, where p, r are both in T and T , and there are no further points of T or T on the dotted lines. The rotation is a right rotation if q is north-east from q, and a left rotation otherwise. p p q q
r
r
Fig. 3. Rotation operation on ν-trees.
Let T be a ν-tree. By connecting each element p of T with the first element
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of T below it (if any), and with the first element of T to its right (if any), we can view T as a rooted binary tree in the graph-theoretical sense; see Figure 1 (middle). This also makes our notion of ν-tree rotation more self-explanatory. Lemma 3.1 A rotation of a ν-tree is also a ν-tree. Strikingly, the complements of ν-trees determine all reduced expressions of πν in Qν [11, Theorem 2.1]. Moreover, two facets are adjacent if and only if the corresponding ν-trees are related by a rotation. Figure 4 illustrates an example. Theorem 2.2 then follows from the following main result. Theorem 3.2 The ν-Tamari lattice is isomorphic to the rotation lattice of ν-trees, whose covering relation is given by right rotation. s2 s2
s4
s4
rotation
s3
s2 s3 s2 s4 = [1, 4, 3, 5, 2, 6]
s2
s3
s3 s3 s2 s3 s4 = [1, 4, 3, 5, 2, 6]
Fig. 4. Complements of ν-trees are the reduced expressions of πν in Qν .
4
A simple proof of the lattice property
The meet and join operations in the ν-Tamari lattice turn out to be very simple when working with ν-trees instead of ν-paths. Definition 4.1 Let ν be a lattice path from (0, 0) to (a, b) with length = (ν) = a + b. The minimal ν-bracket vector bmin is a vector consisting of +1 non-negative integers obtained by reading the y-coordinates of the lattice points on ν in the order they appear along the path. We define the set of fixed positions as the set F = {f0 , f1 , . . . , fb } where fk is the position of the last appearance of k in bmin . A ν-bracket vector is a vector b = (b1 , . . . , b+1 ) satisfying the following properties: (i) bfk = k for 0 ≤ k ≤ b; (ii) bmin ≤ bi ≤ b for all i; i (iii) if bi = k, then bj ≤ k for i ≤ j ≤ fk . Theorem 4.2 The poset of ν-bracket vectors under componentwise order is a lattice isomorphic to the ν-Tamari lattice. The ν-bracket vectors can be obtained from traversals of ν-trees. Recall that the in-order traversal of a binary tree is defined recursively as follows: if
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x is a node with is left subtree A and right subtree B, we traverse the nodes of A in in-order, then x and finally traverse B in in-order. Definition 4.3 We label each node of a ν-tree T by the entry of its ycoordinate. The bracket vector b(T ) is the result of reading the labels of the nodes in in-order. See Figure 1. The meet operation can be easily described in terms of ν-bracket vectors: Proposition 4.4 The meet on the set of ν-bracket vectors is given by componentwise minimum. That is b ∧ b = (min{b1 , b1 }, . . . , min{b+1 , b+1 }).
5
Multi ν-Tamari complexes
For any integer k ≥ 1 one may define the (k, ν)-Tamari complex as the simplicial complex on V whose minimal non-faces are (k + 1)-subsets of pairwise non ν-compatible elements. For ν = (N E)n this coincides with the simplicial complex of (k + 1)-crossing-free subsets of diagonals of a convex (n + 2)-gon, whose facet adjacency graph is conjectured to be realizable as the edge graph of a simple polytope: the multiassociahedron Δn+2,k . For k = 1 we have recently shown that the facet adjacency graph of (1, ν)-Tamari complexes can be realized as the edge graphs of polytopal subdivisions of associahedra 5 [5]. We believe that a similar statement might be true for general k. The following proposition is a first positive result in this direction:
Fig. 5. Facet adjacency graph of the (k, ν)-Tamari complex for k = 2 and ν = (NE5 )3 (left), and k = 3 and ν = (NE5 )4 (right).
Proposition 5.1 Let m ≥ k and ν = (N E m )k+1 . The facet adjacency graph Gk,ν of the Fuss-Catalan (k, ν)-Tamari complex is the edge graph of a polytopal subdivision of the multi-associahedron Δ2k+2,k . More precisely, Δ2k+2,k is a k-dimensional simplex, and Gk,ν is the edge graph of the staircase subdivision of its (m − k + 1) dilation, see Figure 5. 5
Whenever ν does not have two consecutive non-initial north steps
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