Shuffle-compatible permutation statistics II: the exterior peak set
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.
Keywords
Cite
@article{arxiv.1806.04114,
title = {Shuffle-compatible permutation statistics II: the exterior peak set},
author = {Darij Grinberg},
journal= {arXiv preprint arXiv:1806.04114},
year = {2022}
}
Comments
66 pages; a more detailed version is archived as an ancillary file. v3 has been updated, particularly with a new Question 2.51, as well as Remark 2.41 and Example 2.43, as well as fewer errors thanks to a helpful referee