Coxeter-biCatalan combinatorics

TL;DR

This paper introduces Coxeter-biCatalan combinatorics, defining a common counting number for twin pairs across various Coxeter-Catalan structures, and computes these numbers for all finite Coxeter groups.

Contribution

It establishes the concept of W-biCatalan numbers, unifying multiple Coxeter-Catalan counting problems and providing initial computations for all finite Coxeter groups.

Findings

All counting problems have the same solution, the W-biCatalan number.

Computed W-biCatalan numbers for all finite Coxeter groups.

Initiated the study of Coxeter-biCatalan combinatorics.

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.