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Coxeter-biCatalan combinatorics

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Abstract

We pose counting problems related to the various settings for Coxeter-Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is to count “twin” pairs of objects from a corresponding problem in Coxeter-Catalan combinatorics. We show that the problems all have the same answer, and, for a given finite Coxeter group W, we call the common solution to these problems the W-biCatalan number. We compute the W-biCatalan number for all W and take the first steps in the study of Coxeter-biCatalan combinatorics.

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Acknowledgements

Bruno Salvy’s and Paul Zimmermann’s package GFUN [41] was helpful in guessing a formula for the \(D_n\)-Catalan number. John Stembridge’s packages posets and coxeter/weyl [45] were invaluable in counting antichains in the doubled root poset, in checking the distributivity of the doubled root poset, and in verifying the simpliciality of the bipartite biCambrian fan. The authors thank Christos Athanasiadis, Christophe Hohlweg, Richard Stanley, Salvatore Stella, and Bernd Sturmfels for helpful suggestions and questions.

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  1. Department of Mathematics, North Carolina State University, Raleigh, NC, USA

    Emily Barnard & Nathan Reading

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  1. Emily Barnard
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  2. Nathan Reading
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Correspondence to Nathan Reading.

Additional information

Emily Barnard was supported in part by NSF Grants DMS-0943855, DMS-1101568, and DMS-1500949. Nathan Reading was supported in part by NSF Grants DMS-1101568 and DMS-1500949.

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Barnard, E., Reading, N. Coxeter-biCatalan combinatorics. J Algebr Comb 47, 241–300 (2018). https://doi.org/10.1007/s10801-017-0775-1

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