A sequence of quasipolynomials arising from random numerical semigroups

TL;DR

This paper investigates the properties of a sequence of quasipolynomials derived from the expected number of generators in random numerical semigroups, revealing new structural insights.

Contribution

It establishes a recurrence relation proving that the sequence $h_{n,i}$ is eventually quasipolynomial for fixed second parameters.

Findings

The sequence $h_{n,i}$ is eventually quasipolynomial.

A recurrence relation for $h_{n,i}$ is derived.

The expected number of generators can be expressed via this sequence.

Abstract

A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of integers, denoted $h_{n, i}$, that count generating sets with certain properties. We prove a recurrence that implies the sequence $h_{n,i}$ is eventually quasipolynomial when the second parameter is fixed.