Orbit Dirichlet series and multiset permutations

• Published in: Monatshefte für Mathematik Vol. 186; no. 2; pp. 215 - 233
• Main Authors: Carnevale, Angela, Voll, Christopher
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.06.2018; Springer Nature B.V
• Subjects: Combinatorial analysis; Dirichlet problem; Functional equations; Mathematics; Mathematics and Statistics; Permutations; Polynomials; Sequences; Subgroups; 11M41; Orbit Dirichlet series; 30B50; 37P35; Hadamard products of rational generating functions; Eulerian polynomials; Natural boundaries; Igusa functions; 37C30; local functional equations; 05A19; Multiset permutations; 05A15; Carlitz’s
• ISSN: 0026-9255; 1436-5081
• DOI: 10.1007/s00605-017-1128-9

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.