Séminaire Lotharingien de Combinatoire, 93B.59 (2025), 12 pp.
Per Alexandersson and Aryaman Jal
Rook Matroids and Log-Concavity of P-Eulerian Polynomials
Abstract.
We establish the ultra-log-concavity of P-Eulerian polynomials for naturally labeled posets P of width two. This takes a step towards resolving a log-concavity conjecture of Brenti (1989) and completes the story of the Neggers-Stanley conjecture in this special case. We do so by introducing rook matroids, the bases of which are certain restricted rook placements on a skew Ferrers board. The associated generating polynomial of these rook placements is ultra-log-concave. We exhibit a bijection between bases of the rook matroid and linear extensions of a width two poset from which the main result follows. Along the way, we study the structure theory of rook matroids and note that they form a subclass of transversal matroids and positroids. They also enjoy a strong correspondence with lattice path matroids.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.
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