Combinatorics of certain higher $q,t$-Catalan polynomials: chains, joint symmetry, and the Garsia-Haiman formula
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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2012
- DOI:
- 10.48550/arXiv.1211.2191
- arXiv:
- arXiv:1211.2191
- Bibcode:
- 2012arXiv1211.2191L
- Keywords:
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- Mathematics - Combinatorics;
- 05E10 (Primary) 05A19;
- 05E05 (Secondary)
- E-Print:
- In this version we made some minor revisions, including an improvement of the proof of Theorem 2