Coloured shuffle compatibility, Hadamard products, and ask zeta functions

TL;DR

This paper introduces a method to compute Hadamard products of rational generating functions related to ask zeta functions and orbit-counting, utilizing shuffle compatibility of coloured permutation statistics.

Contribution

It extends existing work by developing an explicit approach based on shuffle compatibility for calculating these combinatorial generating functions.

Findings

Provides a new explicit method for Hadamard products of rational functions

Connects shuffle compatibility with computation of ask zeta functions

Extends Gessel and Zhuang's work on coloured permutation statistics

Abstract

We devise an explicit method for computing combinatorial formulae for Hadamard products of certain rational generating functions. The latter arise naturally when studying so-called ask zeta functions of direct sums of modules of matrices or class- and orbit-counting zeta functions of direct products of nilpotent groups. Our method relies on shuffle compatibility of coloured permutation statistics and coloured quasisymmetric functions, extending recent work of Gessel and Zhuang.