A certain sequence on pure $\kappa-$sparse gapsets

TL;DR

This paper investigates the properties of pure κ-sparse gapsets, analyzing a specific integer sequence, and establishes a combinatorial equivalence between certain classes of gapsets based on genus and maximum gap size.

Contribution

It introduces new results on the enumeration of pure κ-sparse gapsets, including a proof of cardinality equivalences and explicit counts for symmetric and pseudo-symmetric cases.

Findings

Cardinality of gapsets with genus 3n+1 and max gap 2n equals that with genus 3n+2 and max gap 2n+1.

Explicit enumeration of symmetric and pseudo-symmetric gapsets in these classes.

Verification of a sequence listed in OEIS as A374773.

Abstract

In this paper, we study the pure $\kappa-$sparse gapsets and our focus on getting information about the sequence observed in Table 3 at [1], this sequence is listed in OEIS as A374773. We verify that the cardinality of the set of gapsets with genus $3n+1$ such that the maximum distance between two consecutive elements is $2n$ is equal to the cardinality of the set of gapsets with genus $3n+2$ such that the maximum distance between two consecutive elements is $2n+1$, for all $n\in \mathbb{N}$. In particular, we compute the cardinality of the symmetric and pseudo-symmetric gapsets in these cases.