Orbit Dirichlet series and multiset permutations

TL;DR

This paper explores the connection between orbit counting Dirichlet series and multiset permutation statistics, providing new functional equations, convergence results, and asymptotic behaviors for these series.

Contribution

It introduces a novel combinatorial interpretation of orbit Dirichlet series via multiset permutation statistics, extending classical polynomial identities and analyzing their analytic properties.

Findings

Established local functional equations for Euler factors.

Determined abscissae of convergence and meromorphic continuation.

Described asymptotics of orbit growth sequences.

Abstract

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz's $q$-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series' abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.