A Combinatorial Approach to the Symmetry of $q,t$-Catalan Numbers

TL;DR

This paper explores the symmetry of $q,t$-Catalan numbers through combinatorial methods, proposing structural decompositions and infinite chains of partitions to explain joint symmetry for specific degrees.

Contribution

It introduces a combinatorial framework with structural decompositions and infinite chains of partitions to explain the joint symmetry of $q,t$-Catalan numbers.

Findings

Proved joint symmetry for degrees up to rac{n}{2}-9 for all n.

Developed a combinatorial approach using mutually opposite subcollections.

Constructed infinite chains of partitions inducing symmetry.

Abstract

The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into "mutually opposite" subcollections that lead to a bijective explanation of joint symmetry in certain cases. A key new idea is the construction of infinite chains of partitions that are independent of $n$ but induce the joint symmetry for all $n$ simultaneously. Using these methods, we prove combinatorially that for $0\leq k\leq 9$ and all $n$, the terms in $C_n(q,t)$ of total degree $\binom{n}{2}-k$ have the required symmetry property.