A Proof of the Symmetric Theta Conjecture when q = 0
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917710479228928 |
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| author | Caraceni, Alessandra Iraci, Alessandro |
| author_facet | Caraceni, Alessandra Iraci, Alessandro |
| contents | In 10.1093/imrn/rnac258, the authors conjecture a combinatorial formula for the expressions $Ξe_α\rvert_{t=1}$, known as Symmetric Theta Trees Conjecture, in terms of tiered trees with an inversion statistic. In 10.1017/fms.2024.14, the authors prove a combinatorial formula for the same symmetric function, in terms of doubly labelled Dyck paths with the area statistic. In this paper, we give an explicit bijection between the subsets of the two families of objects when the relevant statistic is equal to $0$, thus proving the Symmetric Theta Tree Conjecture when $q=0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_02368 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Proof of the Symmetric Theta Conjecture when q = 0 Caraceni, Alessandra Iraci, Alessandro Combinatorics 05E05, 05C05, 05A19 In 10.1093/imrn/rnac258, the authors conjecture a combinatorial formula for the expressions $Ξe_α\rvert_{t=1}$, known as Symmetric Theta Trees Conjecture, in terms of tiered trees with an inversion statistic. In 10.1017/fms.2024.14, the authors prove a combinatorial formula for the same symmetric function, in terms of doubly labelled Dyck paths with the area statistic. In this paper, we give an explicit bijection between the subsets of the two families of objects when the relevant statistic is equal to $0$, thus proving the Symmetric Theta Tree Conjecture when $q=0$. |
| title | A Proof of the Symmetric Theta Conjecture when q = 0 |
| topic | Combinatorics 05E05, 05C05, 05A19 |
| url | https://arxiv.org/abs/2407.02368 |