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arXiv · 1211.2191

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

Kyungyong Lee·Li Li·Nicholas A. Loehr
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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.

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Kyungyong Lee, Li Li, Nicholas A. Loehr. 2012-11-09. Combinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula. https://arxiv.org/abs/1211.2191

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