ALGEBRAIC COMBINATORICS

no. 2

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 σ∈𝔖 n (Π) and Desσ is the descent set of σ. We characterize those Π⊆𝔖 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.

Received: 2018-10-25
Revised: 2019-06-06
Accepted: 2019-09-17
Published online: 2020-04-01
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 = {alco.centre-mersenne.org/item/ALCO_2020__3_2_365_0/}
}
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/item/ALCO_2020__3_2_365_0/

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