From quasisymmetric functions to zeta functions through shuffle compatibility

Abstract: A permutation statistic $st$ is called shuffle compatible if, for any two permutations $\pi$ and $\sigma$ on disjoint sets of symbols, its distribution over all shuffles of $\pi$ and $\sigma$ depends only on $st(\pi)$, $st(\sigma)$ and the lengths of $\pi$ and $\sigma$. It follows from Stanley’s work that the descent set constitutes the prototypical example of a shuffle-compatible permutation statistic. Gessel and Zhuang formalized this notion by introducing and studying the associated shuffle algebra. In this talk, we are going to discuss a colored analogue of shuffle compatibility for colored permutation statistics and its connection with Poirier’s colored quasisymmetric functions. Additionally, we will present an application of colored shuffle compatibility in computing Hadamard products of certain algebraic zeta functions. This is based on joint work with Angela Carnevale and Tobias Rossmann.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar