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arXiv · 2407.21226

Polyhedral geometry of refined $q,t$-Catalan numbers

Matthias Beck·Mitsuki Hanada·Max Hlavacek·John Lentfer·Andrés R. Vindas-Meléndez·Katie Waddle
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Abstract

We study a refinement of the $q,t$-Catalan numbers introduced by Xin and Zhang (2022, 2023) using tools from polyhedral geometry. These refined $q,t$-Catalan numbers depend on a vector of parameters $\vec{k}$ and the classical $q,t$-Catalan numbers are recovered when $\vec{k} = (1,\ldots,1)$. We interpret Xin and Zhang's generating functions by developing polyhedral cones arising from constraints on $\vec{k}$-Dyck paths and their associated area and bounce statistics. Through this polyhedral approach, we recover Xin and Zhang's theorem on $q,t$-symmetry of the refined $q,t$-Catalan numbers in the cases where $\vec{k} = (k_1,k_2,k_3)$ and $(k,k,k,k)$, give some extensions, including the case $\vec{k} = (k,k+m,k+m,k+m)$, and discuss relationships to other generalizations of the $q,t$-Catalan numbers.

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Matthias Beck, Mitsuki Hanada, Max Hlavacek, John Lentfer, Andrés R. Vindas-Meléndez, Katie Waddle. 2024-07-30. Polyhedral geometry of refined $q,t$-Catalan numbers. https://arxiv.org/abs/2407.21226

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