Cyclotomic factors of the descent set polynomial

Authors:
Denis Chebikin;Richard Ehrenborg;Pavlo Pylyavskyy;Margaret Readdy
Affiliations:
Department of Mathematics, MIT, Cambridge, MA 02139, United States;Department of Mathematics, University of Kentucky, Lexington, KY 40506, United States;Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, United States;Department of Mathematics, University of Kentucky, Lexington, KY 40506, United States
Venue:
Journal of Combinatorial Theory Series A
Year:
2009

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Abstract

We introduce the notion of the descent set polynomial as an alternative way of encoding the sizes of descent classes of permutations. 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 in the symmetric group S"n only depends on the number on 1's in the binary expansion of n. We observe similar properties for the signed descent set statistics.