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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.21755 |
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| _version_ | 1866913454157201408 |
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| author | Dey, Hiranya Kishore Mahato, Iswar |
| author_facet | Dey, Hiranya Kishore Mahato, Iswar |
| contents | A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k \geq 1.$ In this paper, we find the sign-balance for mod-k-alternating permutations with respect to the statistic excedance. Moreover, we study the sign-balance for excedances over mod-k-alternating derangements. The results are obtained by constructing suitable matrices and connecting their determinants with the signed excedance enumeration of mod-k-alternating permutations. As an application of the signed excedance enumeration, we prove that when $n \equiv k \pmod {2k}$, the excedance enumerating polynomials over the even and odd mod-k-alternating permutations, starting with a fixed remainder, are gamma-positive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21755 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sign-balance of excedances over mod-k-alternating permutations and gamma-positivity Dey, Hiranya Kishore Mahato, Iswar Combinatorics 05A15, 15A15, 05E99 A permutation is called mod-k-alternating if its entries are restricted to having the same remainder as the index, modulo some integer $k \geq 1.$ In this paper, we find the sign-balance for mod-k-alternating permutations with respect to the statistic excedance. Moreover, we study the sign-balance for excedances over mod-k-alternating derangements. The results are obtained by constructing suitable matrices and connecting their determinants with the signed excedance enumeration of mod-k-alternating permutations. As an application of the signed excedance enumeration, we prove that when $n \equiv k \pmod {2k}$, the excedance enumerating polynomials over the even and odd mod-k-alternating permutations, starting with a fixed remainder, are gamma-positive. |
| title | Sign-balance of excedances over mod-k-alternating permutations and gamma-positivity |
| topic | Combinatorics 05A15, 15A15, 05E99 |
| url | https://arxiv.org/abs/2407.21755 |