Symmetric multisets of permutations

The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer [5]. For which multisubsets B of the symmetric group S-n is the quasisymmetric function

Q(B) = Sigma(pi is an element of B) F-Des((pi),n)

a symmetric function? Here Des(pi) is the descent set of pi and F-Des(pi),F-n is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a useful characterization of these multisets. Using this characterization we prove a conjecture of Elizalde and Roichman from [2]. Two other corollaries are also given. The first is a new and short proof that conjugacy classes are symmetric sets, a well known result first proved by Gessel and Reutenauer [5]. In our second corollary we give a unified explanation that both left and right multiplication of symmetric multisets, by inverse J-classes, is symmetric. The case of right multiplication was first proved by Elizalde and Roichman in [2].

Title	Symmetric multisets of permutations
Creator	Bloom, Jonathan
Publisher	Journal of Combinatorial Theory Series A
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	August 2020
Date Available	2021-05-27
Type	Article
Language	English
Keyword	Quasisymmetric functions
	Multisets of permutations
	Symmetric sets
	Inverse descent classes
	Fine sets
	Symmetric functions
Bibliographic Citation	Bloom, J. (2020 Aug) "Symmetric multisets of permutations." Journal of Combinatorial Theory Series A 174: 105255
Standard Identifier	Handle 10385/bv73c2022
Standard Identifier	DOI 10.1016/j.jcta.2020.105255
Permalink	http://hdl.handle.net/10385/bv73c2022
Rights Statement	In Copyright
Rights Holders	Elsevier Inc.