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26
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 38 (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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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.
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 11 (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 9 (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).
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 7 (4 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.
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 5 (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.
kdistant crossings and nestings of matchings and partitions
 http://arxiv.org/abs/0812.2725 JANG SOO KIM
"... Abstract. We define and consider kdistant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 3 ..."
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Cited by 3 (2 self)
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Abstract. We define and consider kdistant crossings and nestings for matchings and set partitions, which are a variation of crossings and nestings in which the distance between vertices is important. By modifying an involution of Kasraoui and Zeng (Electronic J. Combinatorics 2006, research paper 33), we show that the joint distribution of kdistant crossings and nestings is symmetric. We also study the numbers of kdistant noncrossing matchings and partitions for small k, which are counted by wellknown sequences, as well as the orthogonal polynomials related to kdistant noncrossing matchings and partitions. We extend Chen et al.’s rcrossings and enhanced rcrossings. 1.
ADMISSIBLE SEQUENCES, PREPROJECTIVE MODULES, AND REDUCED WORDS IN THE WEYL GROUP OF A QUIVER
, 2006
"... Abstract. This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shorte ..."
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Abstract. This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)admissible sequence annihilating the module. A (+)admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following application that strengthens known results of Howlett and FominZelevinsky. For any fixed Coxeter element of the Weyl group associated to an indecomposable symmetric generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words.