Colored multiset Eulerian polynomials via products of simplices

This talk explores a family of polynomials arising from descent statistics on colored permutations of multisets. They generalize the colored Eulerian polynomials that appear frequently in algebraic combinatorics and are known to admit desirable distributional properties, including real-rootedness, log-concavity, unimodality and the alternatingly increasing property. In joint work with Bin Han and Liam Solus, we prove that a large class of colored multiset Eulerian polynomials obtains all of the aforementioned distributional properties as well as others, including bi-gamma-positivity. The proof relies on a connection to convex polytopes: colored multiset Eulerian polynomials arise in lattice point enumeration considerations for products of simplices, enabling the use of polyhedral methods to establish our results.