Two classes of posets with real-rooted chain polynomials
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917114631159808 |
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| author | Athanasiadis, Christos A. Douvropoulos, Theo Kalampogia-Evangelinou, Katerina |
| author_facet | Athanasiadis, Christos A. Douvropoulos, Theo Kalampogia-Evangelinou, Katerina |
| contents | The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to finite Coxeter groups, are shown to have this property. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set $\{1, 2,\dots,n\}$ which have ascents at specified positions is shown to be real-rooted, hence log-concave and unimodal, and a good estimate for the location of the peak is deduced. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_04839 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Two classes of posets with real-rooted chain polynomials Athanasiadis, Christos A. Douvropoulos, Theo Kalampogia-Evangelinou, Katerina Combinatorics 05A15, 05E45, 06A07, 26C10 The coefficients of the chain polynomial of a finite poset enumerate chains in the poset by their number of elements. It has been a challenging open problem to determine which posets have real-rooted chain polynomials. Two new classes of posets, namely those of all rank-selected subposets of Cohen-Macaulay simplicial posets and all noncrossing partition lattices associated to finite Coxeter groups, are shown to have this property. The first result generalizes one of Brenti and Welker. As a special case, the descent enumerator of permutations of the set $\{1, 2,\dots,n\}$ which have ascents at specified positions is shown to be real-rooted, hence log-concave and unimodal, and a good estimate for the location of the peak is deduced. |
| title | Two classes of posets with real-rooted chain polynomials |
| topic | Combinatorics 05A15, 05E45, 06A07, 26C10 |
| url | https://arxiv.org/abs/2307.04839 |