Factorization length distribution for affine semigroups II: asymptotic behavior for numerical semigroups with arbitrarily many generators

TL;DR

This paper derives explicit asymptotic formulas for factorization length statistics in numerical semigroups with many generators, using advanced combinatorial, probabilistic, and harmonic analysis tools.

Contribution

It introduces novel methods combining algebraic combinatorics, probability, and harmonic analysis to analyze factorization lengths in numerical semigroups with arbitrary generators.

Findings

Explicit asymptotic expressions for factorization length statistics

Quasipolynomial/quasirational representations in some cases

Demonstrates power of interdisciplinary techniques in semigroup theory

Abstract

For numerical semigroups with a specified list of (not necessarily minimal) generators, we obtain explicit asymptotic expressions, and in some cases quasipolynomial/quasirational representations, for all major factorization length statistics. This involves a variety of tools that are not standard in the subject, such as algebraic combinatorics (Schur polynomials), probability theory (weak convergence of measures, characteristic functions), and harmonic analysis (Fourier transforms of distributions). We provide instructive examples which demonstrate the power and generality of our techniques. We also highlight unexpected consequences in the theory of homogeneous symmetric functions.