A Cyclic Analogue of Stanley's Shuffle Theorem

TL;DR

This paper introduces a cyclic major index for cycle permutations and provides a cyclic analogue of Stanley's Shuffle Theorem, addressing a previously posed open question in enumerative combinatorics.

Contribution

It develops a cyclic analogue of Stanley's Shuffle Theorem using a new cyclic major index, extending the understanding of cyclic shuffles and their descent statistics.

Findings

Defined the cyclic major index for cycle permutations

Derived a bivariate enumerative formula for cyclic shuffles

Answered an open question on cyclic descent enumeration

Abstract

We introduce the cyclic major index of a cycle permutation and give a bivariate analogue of enumerative formula for the cyclic shuffles with a given cyclic descent numbers due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's Shuffle Theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.