Pattern avoidance and quasisymmetric functions
Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 365-388.

Given a set of permutations Π, let 𝔖 n (Π) denote the set of permutations in the symmetric group 𝔖 n that avoid every element of Π in the sense of pattern avoidance. Given a subset S of {1,,n-1}, let F S be the fundamental quasisymmetric function indexed by S. Our object of study is the generating function Q n (Π)=F Desσ where the sum is over all σ𝔖 n (Π) and Desσ is the descent set of σ. We characterize those Π𝔖 3 such that Q n (Π) is symmetric or Schur nonnegative for all n. In the process, we show how each of the resulting Π can be obtained from a theorem or conjecture involving more general sets of patterns. In particular, we prove results concerning symmetries, shuffles, and Knuth classes, as well as pointing out a relationship with the arc permutations of Elizalde and Roichman. Various conjectures and questions are mentioned throughout.

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DOI: 10.5802/alco.96
Classification: 05E05, 05A05
Keywords: Knuth class, pattern avoidance, quasisymmetric function, Schur function, shuffle, symmetric function, Young tableau

Hamaker, Zachary 1; Pawlowski, Brendan 2; Sagan, Bruce E. 3

1 University of Florida Department of Mathematics Gainesville FL 32611-1941, USA
2 University of Southern California Department of Mathematics Los Angeles CA 90089-2532, USA
3 Michigan State University Department of Mathematics East Lansing MI 48824-1027, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Hamaker, Zachary; Pawlowski, Brendan; Sagan, Bruce E. Pattern avoidance and quasisymmetric functions. Algebraic Combinatorics, Volume 3 (2020) no. 2, pp. 365-388. doi : 10.5802/alco.96. https://alco.centre-mersenne.org/articles/10.5802/alco.96/

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