Combinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula

TL;DR

This paper establishes the equivalence of combinatorial and algebraic definitions of higher $q,t$-Catalan polynomials for small n, proves their joint symmetry through a chain decomposition approach, and derives related unimodality and explicit coefficient formulas.

Contribution

It provides the first proof of the equivalence and joint symmetry of higher $q,t$-Catalan polynomials for small n, introducing a novel chain-based approach for symmetry proofs.

Findings

Proved the equivalence of combinatorial and algebraic definitions for n ≤ 4.

Established joint symmetry of the polynomials for n ≤ 4.

Derived explicit formulas and unimodality results for the coefficients.

Abstract

The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This paper proves the equivalence of the two definitions for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and all $n\leq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_n(q,t)$ for all $m\geq 1$ and all $n\leq 4$. We give analogous results for certain rational-slope $q,t$-Catalan polynomials.