Combinatorial study of the q-Catalan triangle and its generalizations

TL;DR

This paper explores the combinatorial properties of the q-Catalan triangle and its generalizations, providing new interpretations and introducing multivariate extensions.

Contribution

It introduces combinatorial interpretations for the q-Catalan triangle, including new families of pattern-avoiding permutations, and presents multivariate generalizations.

Findings

Established combinatorial interpretations via pattern-avoiding permutations, Dyck paths, and triangulations.

Proved the dual recurrence and co-inversion interpretation of the mirror polynomial.

Introduced the q,p-Catalan triangle and multivariate generalizations.

Abstract

We announce a series of results on the combinatorial study of the q-Catalan triangle (C_{n,k}(q)), defined by C_{n,0}(q)=q^{n(n-1)/2} and C_{n,k}(q)=C_{n,k-1}(q)+q^{n-k-1}C_{n-1,k}(q). We establish combinatorial interpretations via a universal combinatorial family of seven components: four families of pattern-avoiding permutations weighted by inversion or co-inversion statistics, Dyck paths, binary words and triangulations. We introduce the mirror polynomial C-tilde_{n,k}(q)=q^{n(n-1)/2}C_{n,k}(q^{-1}), prove its dual recurrence and co-inversion interpretation. The q,p-Catalan triangle and a multivariate generalization opening the way to cyclotomic q-analogues are introduced. Theorems on the q-Catalan triangle via 312-avoiding permutations and the mirror recurrence are proved completely here. This is the first paper of series W0-W5 on classical and q-deformed interpretations of the Catalan triangle.