2024/09/30 by Per Alexandersson, Aryaman Jal, Alexandersson, Per +1 · 5 citations
Mathematics · #05A15 #05A20 #05B35 #06A07 #26C10 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2410.00127
openalex publication_date 2024/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 a subclass of both transversal matroids and positroids; they also bear a subtle relationship to lattice path matroids that centers around not having the quaternary matroid Q6 as a minor. The enumerative and distributional properties of non-nesting rook placements stand in contrast to those 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 WP 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.