q,t-Catalan measures
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916132102864896 |
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| author | Cavey, Ian |
| author_facet | Cavey, Ian |
| contents | We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on $\mathbb{R}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the $q,t$-Catalan numbers. Our main result realizes the $q,t$-Catalan measures as a limit of higher $q,t$-Catalan numbers $C^{(m)}_n(q,t)$ as $m\to\infty$. We also give a geometric interpretation of the $q,t$-Catalan measures. They are the Duistermaat-Heckman measures of the punctual Hilbert schemes parametrizing subschemes of $\mathbb{C}^2$ supported at the origin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07112 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | q,t-Catalan measures Cavey, Ian Combinatorics Algebraic Geometry 05A19 (Primary) 05E14 (Secondary) We introduce the $q,t$-Catalan measures, a sequence of piece-wise polynomial measures on $\mathbb{R}^2$. These measures are defined in terms of suitable area, dinv, and bounce statistics on continuous families of paths in the plane, and have many combinatorial similarities to the $q,t$-Catalan numbers. Our main result realizes the $q,t$-Catalan measures as a limit of higher $q,t$-Catalan numbers $C^{(m)}_n(q,t)$ as $m\to\infty$. We also give a geometric interpretation of the $q,t$-Catalan measures. They are the Duistermaat-Heckman measures of the punctual Hilbert schemes parametrizing subschemes of $\mathbb{C}^2$ supported at the origin. |
| title | q,t-Catalan measures |
| topic | Combinatorics Algebraic Geometry 05A19 (Primary) 05E14 (Secondary) |
| url | https://arxiv.org/abs/2210.07112 |