Cyclotomic factors of the descent set polynomial
Combinatorics
2017-05-30 v2
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.
Cite
@article{arxiv.0705.2451,
title = {Cyclotomic factors of the descent set polynomial},
author = {Denis Chebikin and Richard Ehrenborg and Pavlo Pylyavskyy and Margaret Readdy},
journal= {arXiv preprint arXiv:0705.2451},
year = {2017}
}
Comments
21 pages, revised the proof of the opening result and cleaned up notation