On the number of L-shapes in embedding dimension four numerical semigroups

Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs.This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups SnSn, for odd n≥5n≥5, such that the number of related L-shapes is n+32. This family has her analogue to weighted Cayley digraphs of degree three.Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs.