Revisiting Pattern Avoidance and Quasisymmetric Functions

Let Sn be the nth symmetric group. Given a set of permutations., we denote by Sn(.) the set of permutations in Sn which avoid. in the sense of pattern avoidance. Consider the generating function Qn(.) =s FDes s where the sum is over all s. Sn(.) and FDes s is the fundamental quasisymmetric function corresponding to the descent set of s. Hamaker, Pawlowski, and Sagan introduced Qn(.) and studied its properties, in particular, finding criteria for when this quasisymmetric function is symmetric or even Schur nonnegative for all n = 0. The purpose of this paper is to continue their investigation by answering some of their questions, proving one of their conjectures, as well as considering other natural questions about Qn(.). In particular, we look at. of small cardinality, superstandard hooks, partial shuffles, Knuth classes, and a stability property.

Title	Revisiting Pattern Avoidance and Quasisymmetric Functions
Creator	Bloom, Jonathan
Creator	Sagan, B. E.
Publisher	Annals of Combinatorics
Academic Department	Mathematics
Division	Natural Sciences
Organization	Lafayette College
Date Issued	June 2020
Date Available	2021-05-27
Type	Article
Language	English
Keyword	Pattern avoidance
	Quasisymmetric function
	Schur function
	Shuffle
Bibliographic Citation	Bloom, J. and B. E. Sagan (2020 Jun) "Revisiting Pattern Avoidance and Quasisymmetric Functions." Annals of Combinatorics 24 (2): 337-361.
Standard Identifier	DOI 10.1007/s00026-020-00492-6
Standard Identifier	Handle 10385/nz8061183
Permalink	http://hdl.handle.net/10385/nz8061183
Rights Statement	In Copyright
Rights Holders	Springer Nature