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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2511.05676 |
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| _version_ | 1866909893985828864 |
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| author | Lee, Jeongwon Lesnevich, Nathan Precup, Martha |
| author_facet | Lee, Jeongwon Lesnevich, Nathan Precup, Martha |
| contents | For a finite subset $I$ of positive integers, the descent polynomial $\mathcal{D}(I;n)$ counts the number of permutations in $S_n$ that have descent set $I$. We generalize descent polynomials by considering permutations with a specific subset $S$ of common inversions called $\mathbf{h}$-inversions, where $\mathbf{h} = (\mathbf{h}(1), \mathbf{h}(2), \ldots )$ is a weakly increasing sequence of positive integers such that $\mathbf{h}(i)> i$. We prove that this more general count, denoted by $\mathcal{I}_\mathbf{h}(S;n)$, is also a polynomial. We give three explicit expansions for $\mathcal{I}_\mathbf{h}(S;n)$, prove the coefficients for two of these expansions are log-concave, and define a graded generalization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_05676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Restricted inversion polynomials Lee, Jeongwon Lesnevich, Nathan Precup, Martha Combinatorics 05A05, 05E16 For a finite subset $I$ of positive integers, the descent polynomial $\mathcal{D}(I;n)$ counts the number of permutations in $S_n$ that have descent set $I$. We generalize descent polynomials by considering permutations with a specific subset $S$ of common inversions called $\mathbf{h}$-inversions, where $\mathbf{h} = (\mathbf{h}(1), \mathbf{h}(2), \ldots )$ is a weakly increasing sequence of positive integers such that $\mathbf{h}(i)> i$. We prove that this more general count, denoted by $\mathcal{I}_\mathbf{h}(S;n)$, is also a polynomial. We give three explicit expansions for $\mathcal{I}_\mathbf{h}(S;n)$, prove the coefficients for two of these expansions are log-concave, and define a graded generalization. |
| title | Restricted inversion polynomials |
| topic | Combinatorics 05A05, 05E16 |
| url | https://arxiv.org/abs/2511.05676 |