A conjectured formula for the rational $q,t$-Catalan polynomial
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912170732683264 |
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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 |