Shuffle-compatible permutation statistics. II: The exterior peak set (Q1991426)

Summary: This paper is a continuation of the work of \textit{I. M. Gessel} and \textit{Y. Zhuang} [Adv. Math. 332, 85--141 (2018; Zbl 1388.05008)] (but can be read independently from the latter). We study the shuffle-compatibility of permutation statistics -- a concept introduced by Gessel and Zhuang [loc. cit.], 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.