Cyclic shuffle-compatibility via cyclic shuffle algebras

TL;DR

This paper introduces an algebraic framework for cyclic shuffle-compatibility of permutation statistics, extending existing concepts to cyclic permutations using cyclic quasisymmetric functions, and provides explicit descriptions and proofs for various statistics.

Contribution

It develops a cyclic shuffle algebra framework replacing quasisymmetric functions with cyclic quasisymmetric functions, and applies it to characterize cyclic shuffle-compatibility.

Findings

Defined cyclic shuffle algebra for cyclic shuffle-compatible statistics.

Provided explicit descriptions of cyclic shuffle algebras for various statistics.

Supplied algebraic proofs of cyclic shuffle-compatibility for these statistics.

Abstract

A permutation statistic $\operatorname{st}$ is said to be shuffle-compatible if the distribution of $\operatorname{st}$ over the set of shuffles of two disjoint permutations $\pi$ and $\sigma$ depends only on $\operatorname{st}\pi$, $\operatorname{st}\sigma$, 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 Sietsema 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.