# Counting numerical semigroups by genus and even gaps via Kunz-coordinate vectors

## TL;DR

This paper establishes a bijection between certain numerical semigroups and points in a rational polytope, providing a new approach to analyze the sequence counting semigroups by genus.

## Contribution

It introduces a novel correspondence between numerical semigroups with specific properties and rational polytope points, aiding in understanding their enumeration.

## Key findings

- Provides a method to study the sequence of semigroups by genus
- Offers a geometric perspective via polytope correspondence
- Aims to determine if the sequence of counts is increasing

## Abstract

We contruct a one-to-one correspondence between a subset of numerical semigroups with genus $g$ and $\gamma$ even gaps and the integer points of a rational polytope. In particular, we give an overview to apply this correspondence to try to decide if the sequence $(n_g)$ is increasing, where $n_g$ denotes the number of numerical semigroups with genus $g$.