Journal of Algebraic Combinatorics

, Volume 47, Issue 2, pp 241–300 | Cite as

Coxeter-biCatalan combinatorics



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.


Alternating arc diagram Coxeter-Catalan combinatorics Doubled root poset Twin clusters Twin noncrossing partitions Twin nonnesting partitions Twin sortable elements 



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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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsNorth Carolina State UniversityRaleighUSA

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