# 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/

 Adin, Ron M.; Roichman, Yuval Matrices, characters and descents, Linear Algebra Appl., Volume 469 (2015), pp. 381-418 | Article | MR 3299069 | Zbl 1305.05227

 Bloom, Jonathan; Sagan, Bruce E. Revisiting patterns avoidance and quasisymmetric functions (2018) (preprint, https://arxiv.org/abs/1812.10738)

 Bóna, Miklós Combinatorics of permutations, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004, xiv+383 pages | MR MR2078910 | Zbl 1052.05001

 Elizalde, Sergi; Roichman, Yuval Arc permutations, J. Algebr. Comb., Volume 39 (2014) no. 2, pp. 301-334 | Article | MR 3159254 | Zbl 1292.05268

 Elizalde, Sergi; Roichman, Yuval Signed arc permutations, J. Comb., Volume 6 (2015) no. 1-2, pp. 205-234 | Article | MR 3338850 | Zbl 1312.05007

 Elizalde, Sergi; Roichman, Yuval On rotated Schur-positive sets, J. Comb. Theory, Ser. A, Volume 152 (2017), pp. 121-137 | Article | MR 3682729 | Zbl 1369.05208

 Elizalde, Sergi; Roichman, Yuval Schur-positive sets of permutations via products and grid classes, J. Algebr. Comb., Volume 45 (2017) no. 2, pp. 363-405 | Article | MR 3604061 | Zbl 1362.05128

 Erdős, Paul; Szekeres, George A combinatorial problem in geometry, Compos. Math., Volume 2 (1935), pp. 463-470 | Numdam | MR 1556929 | Zbl 61.0651.04

 Gessel, Ira M. Multipartite $P$-partitions and inner products of skew Schur functions, Combinatorics and algebra (Boulder, Colo., 1983) (Contemp. Math.), Volume 34, Amer. Math. Soc., Providence, RI, 1984, pp. 289-317 | Article | MR MR777705 (86k:05007) | Zbl 0562.05007

 Macdonald, Ian G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, x+475 pages (With contributions by A. Zelevinsky, Oxford Science Publications) | MR MR1354144 (96h:05207) | Zbl 0824.05059

 Malvenuto, Claudia; Reutenauer, Christophe Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, Volume 177 (1995) no. 3, pp. 967-982 | Article | MR 1358493 | Zbl 0838.05100

 Robinson, Gilbert de B. On the Representations of the Symmetric Group, Am. J. Math., Volume 60 (1938) no. 3, pp. 745-760 | Article | MR MR1507943 | Zbl 0019.25102

 Sagan, Bruce E. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, Graduate Texts in Mathematics, 203, Springer-Verlag, New York, 2001, xvi+238 pages | MR MR1824028 (2001m:05261) | Zbl 0964.05070

 Schensted, Craige E. Longest increasing and decreasing subsequences, Can. J. Math., Volume 13 (1961), pp. 179-191 | Article | MR MR0121305 (22 #12047) | Zbl 0097.25202

 Simion, Rodica; Schmidt, Frank W. Restricted permutations, Eur. J. Comb., Volume 6 (1985) no. 4, pp. 383-406 | Article | MR MR829358 (88a:05006) | Zbl 0615.05002

 Stanley, Richard P. Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | MR MR1676282 (2000k:05026) | Zbl 0928.05001

 Stembridge, John R. Enriched $P$-partitions, Trans. Am. Math. Soc., Volume 349 (1997) no. 2, pp. 763-788 | Article | MR 1389788 | Zbl 0863.06005