An area-depth symmetric $q,t$-Catalan polynomial

TL;DR

This paper introduces two symmetric $q,t$-Catalan polynomials based on area, depth, and dinv statistics, proving their symmetry through an involution on plane trees, and connects these results to Tutte polynomials and parking functions.

Contribution

It defines new symmetric $q,t$-Catalan polynomials and proves their symmetry using combinatorial involutions, also relating to Tutte polynomials and graph enumeration.

Findings

Proved symmetry of new $q,t$-Catalan polynomials.

Established involution on plane trees for symmetry proof.

Connected Catalan polynomials to Tutte polynomials and parking functions.

Abstract

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.