Introducing irrational enumeration: analytic combinatorics for objects of irrational size

TL;DR

This paper extends analytic combinatorics to include objects with irrational sizes by using Ribenboim series and generalized Dirichlet series, enabling asymptotic analysis of such classes with diverse applications.

Contribution

It introduces a novel framework for analyzing combinatorial objects with irrational sizes using irrational exponents and singularity analysis, broadening the scope of analytic combinatorics.

Findings

Asymptotic formulas derived for irrational size objects

Applications to tilings, factorizations, lattice walks, and plane trees

Identification of phase transitions in irrational combinatorial classes

Abstract

We extend the scope of analytic combinatorics to classes containing objects that have irrational sizes. The generating function for such a class is a power series that admits irrational exponents (which we call a Ribenboim series). A transformation then yields a generalised Dirichlet series from which the asymptotics of the coefficients can be extracted by singularity analysis using an appropriate Tauberian theorem. In practice, the asymptotics can often be determined directly from the original generating function. We illustrate the technique with a variety of applications, including tilings with tiles of irrational area, ordered integer factorizations, lattice walks enumerated by Euclidean length, and plane trees with vertices of irrational size. We also explore phase transitions in the asymptotics of families of irrational combinatorial classes.