Parity distributions among gaps of free numerical semigroups

TL;DR

This paper investigates the parity distribution of gaps in free numerical semigroups, providing formulas based on Apéry sets and characterizing when odd and even gaps are balanced, especially in special cases like compound and geometric sequences.

Contribution

It extends existing results by deriving explicit formulas for gap parity distributions and characterizes conditions for equality of odd and even gaps in free numerical semigroups.

Findings

At least as many odd gaps as even gaps in free numerical semigroups.

Equality of odd and even gaps occurs when all generators are odd.

Special cases include numerical semigroups generated by compound and geometric sequences.

Abstract

In this paper, we extend recent results about the distribution of even and odd gaps of a numerical semigroup. We find that, for any numerical semigroup, the distribution can be computed in terms of the numbers of or the sums of odd and even elements in a corresponding Ap\'ery set. With free numerical semigroups specifically, we show that there are always at least as many odd gaps as even gaps, with equality precisely when the generating elements are all odd. We then specialize these results to the cases of numerical semigroups generated by compound and geometric sequences.