Cyclotomic factors of the descent set polynomial

Details Cyclotomic factors of the descent set polynomial Cyclotomic factors of the descent set polynomial

2007-05-16

arxiv.org/abs/0705.2451v2

Mathematics

Combinatorics

21 pages, revised the proof of the opening result and cleaned up notation

Scientific paper

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.

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