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Séminaire Lotharingien de Combinatoire, 89B.41 (2023), 12 pp.

# Jinting Liang, Bruce E. Sagan and Yan Zhuang

# Cyclic Shuffle-Compatibility and Cyclic Quasisymmetric Functions

**Abstract.**
A permutation statistic st is said to be shuffle-compatible if the distribution of st over the set of shuffles of two disjoint permutations π and σ depends only on st π, st σ, and the lengths of π and σ. This notion is implicit in Stanley's work on *P*-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility in which quasisymmetric functions play an important role. Later, Domagalski et al. defined a version of shuffle-compatibility for statistics on cyclic permutations. We 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 et al.

Received: November 15, 2022.
Accepted: February 20, 2023.
Final version: April 1, 2023.

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