Coxeter-biCatalan combinatorics
arXiv:1605.03524 · doi:10.1007/s10801-017-0775-1
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.
53 pages, 8 figures. version2: Small expository changes to reflect the fact that "double-positive" Catalan polynomials have already appeared as the local h-polynomials of the positive cluster complex (Athanasiadis-Savvidou). version3: Expository changes reflecting referee suggestions. To appear in JACo