Rook matroids and log-concavity of $P$-Eulerian polynomials
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| Format: | Preprint |
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2024
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| author | Alexandersson, Per Jal, Aryaman |
| author_facet | Alexandersson, Per Jal, Aryaman |
| contents | We define and study rook matroids, the bases of which correspond to non-nesting rook placements on a skew Ferrers board. We show that rook matroids are closed under taking duals and direct sums but not minors. Rook matroids are also a subclass of transversal matroids, positroids, and bear a subtle relationship to lattice path matroids that centers around not having the quaternary matroid $Q_{6}$ as a minor. The enumerative and distributional properties of non-nesting rook placements stand in contrast to that of usual rook placements: the non-nesting rook polynomial is not real-rooted in general, and is instead ultra-log-concave. We leverage this property together with a correspondence between rook placements and linear extensions of a poset to show that if $P$ is a naturally labeled width two poset, then the $P$-Eulerian polynomial $W_{P}$ is ultra-log-concave. This takes an important step towards resolving a log-concavity conjecture of Brenti (1989) and completes the story of the Neggers--Stanley conjecture for naturally labeled width two posets. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00127 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rook matroids and log-concavity of $P$-Eulerian polynomials Alexandersson, Per Jal, Aryaman Combinatorics 05A15, 05A20, 05B35, 06A07, 26C10 We define and study rook matroids, the bases of which correspond to non-nesting rook placements on a skew Ferrers board. We show that rook matroids are closed under taking duals and direct sums but not minors. Rook matroids are also a subclass of transversal matroids, positroids, and bear a subtle relationship to lattice path matroids that centers around not having the quaternary matroid $Q_{6}$ as a minor. The enumerative and distributional properties of non-nesting rook placements stand in contrast to that of usual rook placements: the non-nesting rook polynomial is not real-rooted in general, and is instead ultra-log-concave. We leverage this property together with a correspondence between rook placements and linear extensions of a poset to show that if $P$ is a naturally labeled width two poset, then the $P$-Eulerian polynomial $W_{P}$ is ultra-log-concave. This takes an important step towards resolving a log-concavity conjecture of Brenti (1989) and completes the story of the Neggers--Stanley conjecture for naturally labeled width two posets. |
| title | Rook matroids and log-concavity of $P$-Eulerian polynomials |
| topic | Combinatorics 05A15, 05A20, 05B35, 06A07, 26C10 |
| url | https://arxiv.org/abs/2410.00127 |