Falling stars: a fall-decorated rational shuffle theorem
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915468447580160 |
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| author | Iraci, Alessandro Pagaria, Roberto Paolini, Giovanni |
| author_facet | Iraci, Alessandro Pagaria, Roberto Paolini, Giovanni |
| contents | In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture.
We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects.
We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Falling stars: a fall-decorated rational shuffle theorem Iraci, Alessandro Pagaria, Roberto Paolini, Giovanni Combinatorics 05E05 In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths. |
| title | Falling stars: a fall-decorated rational shuffle theorem |
| topic | Combinatorics 05E05 |
| url | https://arxiv.org/abs/2508.20935 |