Shuffle-compatible permutation statistics II: The exterior peak set

This is a continuation of the work “Shuffle-compatible permutation statistics” by Gessel and Zhuang (but can be read independently from the latter). We study the shuffle-compatibility of permutation statistics-a concept introduced by Gessel and Zhuang, 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.