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

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Abstract

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 C m (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 D b(H) for any finite-dimensional hereditary algebra H.

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Authors and Affiliations

  1. Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, 7491, Trondheim, Norway

    Aslak Bakke Buan & Idun Reiten

  2. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 1J4, Canada

    Hugh Thomas

Authors
  1. Aslak Bakke Buan
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  2. Idun Reiten
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  3. Hugh Thomas
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Correspondence to Hugh Thomas.

Additional information

A. B. Buan, I. Reiten, and H. Thomas were supported by STOR-FORSK Grant 167130 from NFR. A.B.B. and I.R. were supported by Grant 196600 from NFR. H.T. was supported by an NSERC Discovery Grant.

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Buan, A.B., Reiten, I. & Thomas, H. From m-clusters to m-noncrossing partitions via exceptional sequences. Math. Z. 271, 1117–1139 (2012). https://doi.org/10.1007/s00209-011-0906-7

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