Coloured shuffle compatibility, Hadamard products, and ask zeta functions
Angela Carnevale, Vassilis Dionyssis Moustakas, Tobias Rossmann
TL;DR
The paper develops an explicit, combinatorial method to compute Hadamard products of rational generating functions that appear as local factors of ask zeta functions and related zeta problems. Central to the approach is the introduction of shuffle-compatible coloured permutation statistics, particularly the triple $(\mathop{des},\mathop{comaj},\mathbf{col})$, and their embedding into shuffle algebras via coloured quasisymmetric functions. The main theoretical contribution is an elementary, constructive embedding that translates Hadamard products into shuffle-convolution data, enabling closed-form formulae $W_{f,\alpha}^{\varepsilon}$ for a broad class of labelled coloured configurations. The authors apply this framework to obtain explicit Hadamard-product representations for global and local zeta functions (including class- and orbit-counting zeta functions) and related combinatorial objects, thereby bridging algebraic combinatorics and enumerative algebra. These results yield new, concrete interpretations of zeta functions in terms of permutation statistics and provide practical tools for symbolic enumeration across uniform families of algebraic structures.
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.