Ideals of QSym, shuffle-compatibility and exterior peaks

With the concept of a shuffle-compatible permutation statistic (arXiv:1706.00750), Gessel and Zhuang have opened up a new direction in the study of quasisymmetric functions. They have shown that various "descent statistics" (e.g., the descent set, the descent number, the major index, the peak set, the peak number) are shuffle-compatible. Every such statistic leads to an ideal of the algebra QSym, and the quotient algebra is often of interest. We shall discuss one particular statistic -- the "exterior peak set" -- whose shuffle-compatibility we prove (it was left open by Gessel and Zhuang). We then proceed to extend the notion of shuffle-compatibility to a stronger notion, which takes the dendriform structure of QSym into account (and which also holds for the exterior peak set).

(Preprint: http://www.cip.ifi.lmu.de/~grinberg/algebra/gzshuf2.pdf )