Thurs. October 2 at 12:30pm in the Conference Room

September 18, 2014 by Michael Grabchak
Categories: Spring 2022
Richard Ehrenborg  University of Kentuky
Title: The descent set polynomial
Abstract: One usually encodes the number of permutations beta(S) in the symmetric group having descent set S via the Eulerian polynomial, where the number of such permutations is the coefficient of t^|S|. We instead introduce the descent set polynomial where the statistic beta(S) is the exponent of t. Descent set polynomials exhibit interesting factorization patterns. We explore the question of when particular cyclotomic factors divide these polynomials. Especially we consider factors of the form Phi_2, Phi_{2p} where p is a prime and double cyclotomic factors. As an instance we deduce that the proportion of odd entries in the descent set statistics for the symmetric group on n elements only depends on the number of 1’s in the binary expansion of n. This is joint work first with Denis Chebikin, Pavlo Pylyavskyy and Margaret Readdy, and second with Brad Fox.