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

Abstract

The higher $q,t$-Catalan polynomial $C_n^{(m)}(q,t)$ 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 $m\ge 1$ and all $n\le 4$. We also give a bijective proof of the joint symmetry property $C_n^{(m)}(q,t)=C_n^{(m)}(t,q)$ for all $m\ge 1$ and all $n\le 4$. 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 $C_n^{(m)}(q,t)$ for all $m\ge 1$ and all $n\le 4$. We give analogous results for certain rational-slope $q,t$-Catalan polynomials.

In this version we made some minor revisions, including an improvement of the proof of Theorem 2

Cited by in corpus (1)

• Compactified Jacobians and q,t-Catalan Numbers, I