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20
Noncrossing partitions for the group Dn
 SIAM J. Discrete Math
"... Dedicated to the memory of Rodica Simion Abstract. The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a selfdual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,..., n} defined by Kreweras in 1972 when W is th ..."
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Cited by 40 (4 self)
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Dedicated to the memory of Rodica Simion Abstract. The poset of noncrossing partitions can be naturally defined for any finite Coxeter group W. It is a selfdual, graded lattice which reduces to the classical lattice of noncrossing partitions of {1, 2,..., n} defined by Kreweras in 1972 when W is the symmetric group Sn, and to its type B analogue defined by the second author in 1997 when W is the hyperoctahedral group. We give a combinatorial description of this lattice in terms of noncrossing planar graphs in the case of the Coxeter group of type Dn, thus answering a question of Bessis. Using this description, we compute a number of fundamental enumerative invariants of this lattice, such as the rank sizes, number of maximal chains and Möbius function. We also extend to the type D case the statement that noncrossing partitions are equidistributed to nonnesting partitions by block sizes, previously known for types A, B and C. This leads to a (casebycase) proof of a theorem valid for all root systems: the noncrossing and nonnesting subspaces within the intersection lattice of the Coxeter hyperplane arrangement have the same distribution according to Worbits. 1. Introduction and
Cambrian Fans
"... Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc betwee ..."
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Cited by 19 (5 self)
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Abstract. For a finite Coxeter group W and a Coxeter element c of W, the cCambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W. Its maximal cones are naturally indexed by the csortable elements of W. The main result of this paper is that the known bijection clc between csortable elements and cclusters induces a combinatorial isomorphism of fans. In particular, the cCambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W. The rays of the cCambrian fan are generated by certain vectors in the Worbit of the fundamental weights, while the rays of the ccluster fan are generated by certain roots. For particular (“bipartite”) choices of c, we show that the cCambrian fan is linearly isomorphic to the ccluster fan. We characterize, in terms of the combinatorics of clusters, the partial order induced, via the map clc, on cclusters by the cCambrian lattice. We give a simple bijection from cclusters to cnoncrossing partitions that respects the refined (Narayana) enumeration. We relate the Cambrian fan to well known objects in the theory of cluster algebras, providing a geometric
Shellability of noncrossing partition lattices
 Proc. Amer. Math. Soc
"... Abstract. We give a casefree proof that the lattice of noncrossing partitions associated to any finite real reflection group is ELshellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three. 1. ..."
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Cited by 17 (3 self)
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Abstract. We give a casefree proof that the lattice of noncrossing partitions associated to any finite real reflection group is ELshellable. Shellability of these lattices was open for the groups of type Dn and those of exceptional type and rank at least three. 1.
Linked Partitions and Linked Cycles
"... The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the nth large Schröder number rn, which counts the number of Schröder paths. In this paper we give a bi ..."
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Cited by 10 (5 self)
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The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the nth large Schröder number rn, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, kStirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs.
Cyclotomic BirmanWenzlMurakami algebras I: Freeness and realization as tangle algebras, preprint
, 2006
"... Abstract. The cyclotomic BirmanWenzlMurakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study the representation theory of these algebras in the generic semisimple case, and admissibility conditions on the ground ring. Conten ..."
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Cited by 9 (5 self)
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Abstract. The cyclotomic BirmanWenzlMurakami algebras are quotients of the affine BMW algebras in which the affine generator satisfies a polynomial relation. We study the representation theory of these algebras in the generic semisimple case, and admissibility conditions on the ground ring. Contents
Extensions of the linear bound in the FürediHajnal conjecture
"... We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1entries in an n × n (0,1)matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bo ..."
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Cited by 8 (1 self)
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We present two extensions of the linear bound, due to Marcus and Tardos, on the number of 1entries in an n × n (0,1)matrix avoiding a fixed permutation matrix. We first extend the linear bound to hypergraphs with ordered vertex sets and, using previous results of Klazar, we prove an exponential bound on the number of hypergraphs on n vertices which avoid a fixed permutation. This, in turn, solves various conjectures of Klazar as well as a conjecture of Brändén and Mansour. We then extend the original Füredi–Hajnal problem from ordinary matrices to ddimensional matrices and show that the number of 1entries in a ddimensional (0,1)matrix with side length n which avoids a ddimensional permutation matrix is O(n d−1).
Catalans intervals and realizers of triangulations
 In Proc. FPSAC07
, 2007
"... The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley or ..."
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Cited by 4 (0 self)
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The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size n as the relation of being above. Hence, intervals in the Stanley lattice are pairs of noncrossing Dyck paths. In a former article, the second author defined a bijection Φ between pairs of noncrossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection Φ. Then, we study the restriction of Φ to Tamari’s and Kreweras ’ intervals. We prove that Φ induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that Φ induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, Φ induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees. 1
Chains in the noncrossing partition lattice
 SIAM J. Discrete Math
"... Abstract. We prove a general recursive formula which counts certain chains in the noncrossing partition lattice of a finite Coxeter group. Using basic facts about noncrossing partitions, the formula is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups). We solve ..."
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Cited by 4 (1 self)
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Abstract. We prove a general recursive formula which counts certain chains in the noncrossing partition lattice of a finite Coxeter group. Using basic facts about noncrossing partitions, the formula is proven uniformly (i.e. without appealing to the classification of finite Coxeter groups). We solve various specializations of the recursion for each finite Coxeter group in the classification. Among other results, we obtain a simpler proof of a known uniform formula for the number of maximal chains of noncrossing partitions and a new uniform formula for the number of edges in the noncrossing partition lattice. All of our results extend to the mdivisible noncrossing partition lattice. 1.
Preprojective representations of valued quivers and reduced words in the Weyl group of a KacMoody algebra
, 2006
"... Abstract. This paper studies connections between the preprojective representations of a valued quiver, the (+)admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)admissible sequence. A (+)admissible sequence is the canonical sequ ..."
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Cited by 3 (1 self)
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Abstract. This paper studies connections between the preprojective representations of a valued quiver, the (+)admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)admissible sequence. A (+)admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett, FominZelevinsky, and the authors.