Aryaman Jal: Rook matroids and log-concavity of P-Eulerian polynomials

Abstract: In 1972,while investigating the theory of monomer-dimer systems, Heilmann and Lieb proved that the matching polynomial of a graph is real-rooted. This seminal result spurred the application of the geometry of polynomials to algebraic combinatorics. In the spirit of the Heilmann-Lieb theorem, we consider the set of non-nesting rook placements on a skew Ferrers board and probe the distributional properties of its generating polynomial. Surprisingly, the answers are governed by a new matroidal structure that we dub the rook matroid. We will discuss its structural properties and focus on its relation to lattice path matroids. We also consider a poset-theoretic perspective of this problem and in doing so, make progress on a conjecture of Brenti (1989) on the log-concavity of P-Eulerian polynomials. In particular, this completes the story of the Neggers-Stanley conjecture in the naturally labeled width two case. This is joint work with Per Alexandersson.