Cyclic shuffle-compatibility via cyclic shuffle algebras

DOI10.1007/S00026-023-00669-9arXiv2212.14522MaRDI QIDQ6422042

Yan Zhuang, Jinting Liang, Bruce E. Sagan

Publication date: 29 December 2022

Abstract: A permutation statistic $operatornamest$ is said to be shuffle-compatible if the distribution of $operatornamest$ over the set of shuffles of two disjoint permutations $pi$ and $sigma$ depends only on $operatornamestpi$, $operatornamestsigma$, and the lengths of $pi$ and $sigma$. Shuffle-compatibility is implicit in Stanley's early work on $P$-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsma defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.

Mathematics Subject Classification ID

Permutations, words, matrices (05A05) Symmetric functions and generalizations (05E05) Combinatorial aspects of representation theory (05E10)

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