Mathematics Colloquia and Seminars

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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 answer is governed by a new matroidal structure that we dub the rook matroid. We will discuss some of the structural properties of the rook matroid in relation to transversal matroids and positroids. 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. This completes the story of the Neggers-Stanley conjecture in the case of naturally labeled posets of width two. This is joint work with Per Alexandersson.