Lattice paths with given number of turns and semimodules over numerical semigroups

TL;DR

This paper explores the relationship between numerical semigroup semimodules and lattice paths, analyzing their classifications, properties, and syzygies, with explicit counts of certain isomorphic classes.

Contribution

It establishes new connections between \

Findings

Count of semimodules isomorphic to their k-th syzygy

Classification of semimodules via lattice paths

Analysis of gaps in numerical semigroups

Abstract

Let \Gamma=<\alpha, \beta > be a numerical semigroup. In this article we consider several relations between the so-called \Gamma-semimodules and lattice paths from (0,\alpha) to (\beta,0): we investigate isomorphism classes of \Gamma-semimodules as well as certain subsets of the set of gaps of \Gamma, and finally syzygies of \Gamma-semimodules. In particular we compute the number of \Gamma-semimodules which are isomorphic with their k-th syzygy for some k.