Parity distributions among gaps of free numerical semigroups
Caleb McKinley Shor
TL;DR
This work generalizes the parity analysis of gaps in numerical semigroups by deriving a general formula for $O(G(S))-E(G(S))$ in terms of Apéry sets relative to any nonzero element and then specializes to free numerical semigroups, where a closed-form expression is obtained in terms of the telescopic data. The main result for free semigroups shows there are always at least as many odd gaps as even gaps, with equality if and only if all minimal generators are odd. The authors further apply these results to compound and geometric sequences, yielding explicit product formulas for the gap parity difference and providing concrete examples, thereby enriching the understanding of gap structure in a broad class of semigroups and suggesting avenues for modulo-based extensions and further study.
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 Apéry 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 sequences.