# ALGEBRAIC COMBINATORICS

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

Given a set of permutations $\Pi$, let ${𝔖}_{n}\left(\Pi \right)$ denote the set of permutations in the symmetric group ${𝔖}_{n}$ that avoid every element of $\Pi$ in the sense of pattern avoidance. Given a subset $S$ of $\left\{1,\cdots ,n-1\right\}$, let ${F}_{S}$ be the fundamental quasisymmetric function indexed by $S$. Our object of study is the generating function ${Q}_{n}\left(\Pi \right)=\sum {F}_{\text{Des}\sigma }$ where the sum is over all $\sigma \in {𝔖}_{n}\left(\Pi \right)$ and $\text{Des}\sigma$ is the descent set of $\sigma$. We characterize those $\Pi \subseteq {𝔖}_{3}$ such that ${Q}_{n}\left(\Pi \right)$ is symmetric or Schur nonnegative for all $n$. In the process, we show how each of the resulting $\Pi$ 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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.96
Classification: 05E05,  05A05
Keywords: Knuth class, pattern avoidance, quasisymmetric function, Schur function, shuffle, symmetric function, Young tableau
@article{ALCO_2020__3_2_365_0,
author = {Hamaker, Zachary and Pawlowski, Brendan and Sagan, Bruce E.},
title = {Pattern avoidance and quasisymmetric functions},
journal = {Algebraic Combinatorics},
pages = {365--388},
publisher = {MathOA foundation},
volume = {3},
number = {2},
year = {2020},
doi = {10.5802/alco.96},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.96/}
}
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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