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

Lee, Kyungyong ; Li, Li ; Loehr, Nicholas A.

Journal of Algebraic Combinatorics, Tome 39 (2014) no. 4, pp. 749-781.

Résumé

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.

Détail

Classification : 05A10, 05A15, 05E05
Keywords: $q, t$-Catalan polynomials, joint symmetry, lattice paths