A Counter Example to the Shuffle Compatiblity Conjecture

TL;DR

This paper presents a counter-example to Gessel and Zhuang's conjecture that all shuffle-compatible permutation statistics are descent statistics, challenging a previously held belief in the field.

Contribution

The paper provides the first known counter-example to the conjecture that all shuffle-compatible permutation statistics are descent statistics.

Findings

Counter-example disproves the conjecture

Not all shuffle-compatible statistics are descent statistics

Challenges existing assumptions in permutation statistic theory

Abstract

The shuffle product has a connection with several useful permutation statistics such as descent and peak, and corresponds to the multiplication operation in the corresponding descent and peak algebras. In their recent work, Gessel and Zhuang formalized the notion of shuffle-compatibility and studied various permutation statistics from this viewpoint. They further conjectured that any shuffle compatible permutation statistic is a descent statistic. In this note we construct a counter-example to this conjecture.