A conjectured formula for the rational $q,t$-Catalan polynomial

Fuente: arXiv
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Main Author: Hawkes, Graham
Format: Preprint
Published: 2022
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author Hawkes, Graham
author_facet Hawkes, Graham
contents We conjecture a formula for the rational $q,t$-Catalan polynomial $\mathcal{C}_{r/s}$ that is symmetric in $q$ and $t$ by definition. The conjecture posits that $\mathcal{C}_{r/s}$ can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite $d^*$, giving a combinatorial proof of our conjecture on the infinite set of functions $\{ \mathcal{C}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \leq d^*\}$ is equivalent to a finite counting problem.
format Preprint
id arxiv_https___arxiv_org_abs_2208_00577
institution arXiv
publishDate 2022
record_format arxiv
spellingShingle A conjectured formula for the rational $q,t$-Catalan polynomial
Hawkes, Graham
Combinatorics
We conjecture a formula for the rational $q,t$-Catalan polynomial $\mathcal{C}_{r/s}$ that is symmetric in $q$ and $t$ by definition. The conjecture posits that $\mathcal{C}_{r/s}$ can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite $d^*$, giving a combinatorial proof of our conjecture on the infinite set of functions $\{ \mathcal{C}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \leq d^*\}$ is equivalent to a finite counting problem.
title A conjectured formula for the rational $q,t$-Catalan polynomial
topic Combinatorics
url https://arxiv.org/abs/2208.00577