Danai Deligeorgaki: Colored multiset Eulerian polynomials

ABSTRACT: The central objects in this talk are the descent polynomials of colored permutations on multisets, referred to as colored multiset Eulerian polynomials. These polynomials 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, symmetric colored multiset Eulerian polynomials are identified and used to prove sufficient conditions for a colored multiset Eulerian polynomial to be interlaced by its own reciprocal. This property implies that the polynomial obtains all of the aforementioned distributional properties as well as others, including bi-gamma-positivity. To derive these results, multivariate generalizations of a generating function identity due to MacMahon are deduced. We will see some of these identities. At the last part of the talk, I would like to discuss applications to special families of polytopes, namely s-lecture hall polytopes, as well as some further connections between posets and multiset permutations that we are exploring together with Matthias Beck.