"The descent set polynomial"

Richard Ehrenborg
University of Kentucky

Monday, February 23, 2009
4:00 pm, 845 Patterson Office Tower

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. 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. Similar properties hold for the signed descent set statistics.

This is joint work with Denis Chebikin, Pavlo Pylyavskyy and Margaret Readdy