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 Liam Solus and Bin Han, 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 end with open questions and some connections between colored multiset Eulerian polynomials and other families of well-studied polynomials, including the \$s\$-Eulerian polynomials.
Note: This talk begins with a pre-seminar (aimed at graduate students) at 3:30–4:00. The main talk starts at 4:10.
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Meeting ID: 915 4733 5974