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From m-clusters to m-noncrossing partitions via exceptional sequences

By Aslak Bakke Buan, Idun Reiten and Hugh Thomas


Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss–Catalan number Cm(W). We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category Db(H) for any finite-dimensional hereditary algebra H

Topics: exceptional objects, noncrossing partitions, clusters, derived categories, generalized Catalan numbers, hereditary algebras.
Year: 2012
DOI identifier: 10.1007/s00209-011-0906-7
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