Semigroups with fixed multiplicity and embedding dimension

Given $m\in \mathbb{N},$ a numerical semigroup with multiplicity $m$ is called packed numerical semigroup if its minimal generating set is included in $\{m,m+1,\dots ,2m-1\}\mathrm{.}$ In this work, packed numerical semigroups are used to built the set of numerical semigroups with fixed multiplicity and embedding dimension, and to create a partition in this set. Moreover, Wilf’s conjecture is checked in the tree associated to some packed numerical semigroups.