Rook matroids and log-concavity of $P$-Eulerian polynomials
| Autor: | Alexandersson, Per, Jal, Aryaman |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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, direct sums but not minors. Rook matroids are also transversal, 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. Comment: 47 pages, 21 figures |
| Databáze: | arXiv |
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