A counterexample to the shuffle compatiblity conjecture (Q2088690)

Summary: The shuffle product has a connection with several useful permutation statistics such as descent and peak, and corresponds to the multiplication operation in the corresponding descent and peak algebras. \textit{I. M. Gessel} and \textit{Y. Zhuang} [Adv. Math. 332, 85--141 (2018; Zbl 1388.05008)] formalized the notion of shuffle-compatibility and studied various permutation statistics from this viewpoint. They further conjectured that any shuffle compatible permutation statistic is a descent statistic. In this note we construct a counter-example to this conjecture.