Polyhedral geometry of refined $q,t$-Catalan numbers
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arXiv
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| Format: | Preprint |
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2024
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| author | Beck, Matthias Hanada, Mitsuki Hlavacek, Max Lentfer, John Vindas-Meléndez, Andrés R. Waddle, Katie |
| author_facet | Beck, Matthias Hanada, Mitsuki Hlavacek, Max Lentfer, John Vindas-Meléndez, Andrés R. Waddle, Katie |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21226 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polyhedral geometry of refined $q,t$-Catalan numbers Beck, Matthias Hanada, Mitsuki Hlavacek, Max Lentfer, John Vindas-Meléndez, Andrés R. Waddle, Katie Combinatorics 05A15 (Primary), 52B20 (Secondary) 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. |
| title | Polyhedral geometry of refined $q,t$-Catalan numbers |
| topic | Combinatorics 05A15 (Primary), 52B20 (Secondary) |
| url | https://arxiv.org/abs/2407.21226 |