Rook matroids and log-concavity of $P$-Eulerian polynomials

TL;DR

The paper develops a rich connection between non-nesting rook placements on skew Ferrers boards and matroid theory, defining rook matroids which are transversal and positroids and exploring their relation to lattice path matroids via 332/1-avoidance. It shows the non-nesting rook polynomial is not generally real-rooted but is ultra-log-concave, and, through a poset-linear-extension correspondence, proves ultra-log-concavity for width-two P-Eulerian polynomials, addressing the width-two case of the Neggers–Stanley conjecture. A detailed study connects rook matroids to lattice path matroids, including a precise obstruction (the $Q_6$ minor) and a spine-path bijection that yields a matroid isomorphism when $332/1$ is avoided. The work further develops a multivariate Lorentzian framework that lifts P-Eulerian polynomials to a Lorentzian object, linking to rook polynomials and establishing HPP-related results for certain lattice path subclasses (snakes/panhandles) while identifying counterexamples in general. Overall, the paper advances the understanding of distributional properties in rook theory, matroid theory, and poset combinatorics, and provides a unifying Lorentzian perspective on P-Eulerian polynomials in width-two cases with implications for Brenti’s log-concavity conjecture.

Abstract

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.

Figures (21)

• Figure 1: Various matroid subclasses.
• Figure 2: Board which does not admit a rook matroid structure, and the seven additional non-nesting rook placements. This does not form a matroid.
• Figure 3: The direct sum $\lambda_{1}/\mu_{1} \oplus \lambda_{2}/\mu_{2}$ of the rook matroids on $\lambda_{1}/\mu_{1} = 54$ and $\lambda_{2}/\mu_{2} = 22$ is the rook matroid on the direct sum of their shapes, $\lambda / \mu = 7754/55$.
• Figure 4: Duals of rook matroids on skew shapes are rook matroids on conjugates of shapes.
• Figure 5: The matroid $\mathcal{R}_{3322/22} \setminus 6 \cong \mathcal{P}_{332/1}$ is not a rook matroid.

Theorems & Definitions (103)

• Example 2.1
• Definition 2.2
• Example 2.3
• Example 2.4: Narayana polynomials
• Definition 2.5
• Theorem 2.6
• Definition 2.7
• Definition 2.8
• Definition 2.9
• Definition 2.10