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17
Generalized cluster complexes and Coxeter combinatorics
 Int. Math. Res. Notices
"... and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial) ..."
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and study a simplicial complex ∆m (Φ) associated to a finite root system Φ and a nonnegative integer parameter m. Form = 1, our construction specializes to the (simplicial)
A geometric description of mcluster categories
"... Abstract. We show that the mcluster category of type An−1 is equivalent to ..."
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Abstract. We show that the mcluster category of type An−1 is equivalent to
The cyclic sieving phenomenon for faces of generalized cluster complexes, arXiv preprint math.CO/0612679
"... Abstract. The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge’s q = −1 phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin ..."
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Abstract. The notion of cyclic sieving phenomenon is introduced by Reiner, Stanton, and White as a generalization of Stembridge’s q = −1 phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type An, Bn, Dn, and I2(a) are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type E6, E7, E8, F4, H3, and H4, a verification for such a phenomenon on their maximal faces is given. 1.
On the enumeration of positive cells in generalized cluster complexes and Catalan . . .
, 2006
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Decomposition numbers for finite Coxeter groups and generalised noncrossing partitions
, 2007
"... Given a finite irreducible Coxeter group W, a positive integer d, and types T1, T2,..., Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ1σ2 · · · σd of a Coxeter element c of W, such that σi is a Coxeter element in a subgroup of type T ..."
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Given a finite irreducible Coxeter group W, a positive integer d, and types T1, T2,..., Td (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions c = σ1σ2 · · · σd of a Coxeter element c of W, such that σi is a Coxeter element in a subgroup of type Ti in W, i = 1, 2,..., d, and such that the factorisation is “minimal ” in the sense that the sum of the ranks of the Ti’s, i = 1, 2,..., d, equals the rank of W. For the exceptional types, these decomposition numbers have been computed by the first author in [“Topics in Discrete
Faces of generalized cluster complexes and noncrossing partitions, preprint
"... Abstract. Let Φ be an irreducible finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆m (Φ) defined by S. Fomin and N. Reading and the poset NC (m) W of mdivisible noncrossing partitions defined by D. Armstrong. W ..."
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Abstract. Let Φ be an irreducible finite root system with corresponding reflection group W and let m be a nonnegative integer. We consider the generalized cluster complex ∆m (Φ) defined by S. Fomin and N. Reading and the poset NC (m) W of mdivisible noncrossing partitions defined by D. Armstrong. We give a characterization of the faces of ∆m (Φ) in terms of NC (m) W, generalizing that of T. Brady and C. Watt given in the case m = 1. Making use of this, we give a case free proof of the “F = M conjecture ” of D. Armstrong, which relates a certain refined face count of ∆m (Φ) with the Möbius function of NC (m)
Shellability and higher CohenMacaulay conectivity of generalized cluster complexes
, 2006
"... Abstract. Let Φ be a finite root system of rank n and let m be a positive integer. It is proved that the generalized cluster complex ∆m (Φ), introduced by S. Fomin and N. Reading, is (m + 1)CohenMacaulay, in the sense of Baclawski. This statement was conjectured by V. Reiner. More precisely, it is ..."
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Abstract. Let Φ be a finite root system of rank n and let m be a positive integer. It is proved that the generalized cluster complex ∆m (Φ), introduced by S. Fomin and N. Reading, is (m + 1)CohenMacaulay, in the sense of Baclawski. This statement was conjectured by V. Reiner. More precisely, it is proved that the simplicial complex obtained from ∆m (Φ) by removing any subset of its vertex set of cardinality not exceeding m is pure, of the same dimension as ∆m (Φ), and shellable. An analogous statement is shown to hold for the positive part ∆m + (Φ) of ∆m (Φ). Finally, an explicit homotopy equivalence is given between ∆m +(Φ) and the poset of generalized noncrossing partitions, associated to the pair (Φ, m) by D. Armstrong. 1.
Dissections, Homcomplexes and the Cayley trick
 J. Combinatorial Theory, Ser. A
, 2005
"... Abstract. We show that certain canonical realizations of the complexes Hom(G, H) and Hom+(G, H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these comple ..."
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Abstract. We show that certain canonical realizations of the complexes Hom(G, H) and Hom+(G, H) of (partial) graph homomorphisms studied by Babson and Kozlov are in fact instances of the polyhedral Cayley trick. For G a complete graph, we then characterize when a canonical projection of these complexes is itself again a complex, and exhibit several wellknown objects that arise as cells or subcomplexes of such projected Homcomplexes: the dissections of a convex polygon into kgons, Postnikov’s generalized permutohedra, staircase triangulations, the complex dual to the lower faces of a cyclic polytope, and the graph of weak compositions of an integer into a fixed number of summands. 1.
complexes and Catalan
"... On the enumeration of positive cells in generalized cluster ..."
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The Rotation Graph of kary Trees is Hamiltonian
, 2006
"... In this paper we show that the graph of kary trees, connected by rotations, contains a Hamilton cycle. Our proof is constructive and thus provides a cyclic Gray code for kary trees. Furthermore, we identify a basic building block of this graph as the 1skeleton of the polytopal complex dual to the ..."
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In this paper we show that the graph of kary trees, connected by rotations, contains a Hamilton cycle. Our proof is constructive and thus provides a cyclic Gray code for kary trees. Furthermore, we identify a basic building block of this graph as the 1skeleton of the polytopal complex dual to the lower faces of a certain cyclic polytope.