$q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements

TL;DR

This paper explores $q$-analogues of super Catalan numbers, deriving identities, generating functions, and Narayana refinements, and proves their algebraic and combinatorial positivity properties.

Contribution

It extends known combinatorial identities to $q$-super Catalan numbers and introduces Narayana-type refinements with proven positivity and $q$-analog properties.

Findings

Derived combinatorial identities for $q$-super Catalan numbers

Established generating functions and $q$-convolution identities

Proved $ ext{γ}$-positivity and positivity of $q$-refinements

Abstract

We begin by deriving a number of combinatorial identities satisfied by the $q$-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the $q$-super Catalan numbers. Next, we introduce some $q$-convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for $q$-Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the $\gamma$-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural $q$-analogs, prove their $q$-$\gamma$-positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in $q$.