Shuffle-compatible permutation statistics II: the exterior peak set

TL;DR

This paper proves the shuffle-compatibility of the exterior peak set statistic and introduces the stronger LR-shuffle-compatibility, connecting these concepts with quasisymmetric functions and their dendriform structure.

Contribution

It establishes the shuffle-compatibility of Epk and introduces the new concept of LR-shuffle-compatibility, expanding understanding of permutation statistics.

Findings

Epk is shuffle-compatible, confirming Gessel and Zhuang's conjecture.

Epk and other statistics are shown to be LR-shuffle-compatible.

Connections are made between shuffle-compatibility and quasisymmetric functions' dendriform structure.

Abstract

This is a continuation of arXiv:1706.00750 by Gessel and Zhuang (but can be read independently from the latter). We study the shuffle-compatibility of permutation statistics -- a concept introduced in arXiv:1706.00750, although various instances of it have appeared throughout the literature before. We prove that (as Gessel and Zhuang have conjectured) the exterior peak set statistic (Epk) is shuffle-compatible. We furthermore introduce the concept of an "LR-shuffle-compatible" statistic, which is stronger than shuffle-compatibility. We prove that Epk and a few other statistics are LR-shuffle-compatible. Furthermore, we connect these concepts with the quasisymmetric functions, in particular the dendriform structure on them.