Séminaire Lotharingien de Combinatoire, B85d (2021), 11 pp.
Rachel Domagalski, Jinting Liang, Quinn Minnich, Bruce E. Sagan, Jamie Schmidt
and Alexander Sietsema
Cyclic Shuffle Compatibility
Abstract.
Consider a permutation π to be any finite list of distinct
positive integers.
A statistic is a function St whose domain is all permutations.
Let (π shuffle σ) be the set of shuffles of two disjoint permutations
π and &si;.
We say that St is shuffle compatible if the distribution of St
over (π shuffle 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.
One of the places where shuffles are useful is in describing the
product in the algebra of quasisymmetric functions. Recently Adin,
Gessel, Reiner, and Roichman defined an algebra of cyclic
quasisymmetric functions where a cyclic version of shuffling comes
into play. The purpose of this paper is to define and study cyclic
shuffle compatibility. In particular, we show how one can lift
shuffle compatibility results for (linear) permutations to cyclic
ones. We then apply this result to cyclic descents and cyclic peaks.
We also discuss the problem of finding a cyclic analogue of the major
index.
Received: June 18, 2021.
Accepted: September 14, 2021.
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