Pattern avoidance and quasisymmetric functions

Hamaker, Zachary¹ ; Pawlowski, Brendan² ; 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

Algebraic Combinatorics, Tome 3 (2020) no. 2, pp. 365-388.

Résumé

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 $\{1,\cdots ,n-1\}$, 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.

Détail

Reçu le : 2018-10-26
Révisé le : 2019-06-07
Accepté le : 2019-09-18
Publié le : 2020-04-01

DOI : 10.5802/alco.96

Classification : 05E05, 05A05
Mots-clés : Knuth class, pattern avoidance, quasisymmetric function, Schur function, shuffle, symmetric function, Young tableau

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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