Shuffle-compatible permutation statistics II: the exterior peak set
arXiv:1806.04114 · doi:10.37236/7946
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.
66 pages; a more detailed version is archived as an ancillary file. v4 fixes several minor errors, particularly in the detailed version (where Theorems 6.1 and 6.12 were missing assumptions)