In this study, the set of all telescopic numerical semigroups families with embedding dimension three will be obtained for some fixed multiplicity by some parameters. Also, some invariants of these families will be calculated in term of their generators

In this work, we give parametrizations of telescopic numerical semigroups with multiplicity ten and embedding dimension three. We also express some of its invariants in terms of generators of these semigroups such as the Frobenius number, genus and Sylvester number.

In this study, we characterize all families of saturated numerical semigroups with multiplicity four. We also present some results about invariants of these semigroups.

In this study, we deal with the concept of special gap of a numerical semigroup which is used to find the set of all numerical semigroups containing a given numerical semigroup. We will find the specific gaps of Arf numerical semigroups with small multiplicity. We also find all Arf numerical semigroups containing a given Arf numerical semigroup with small multiplicity.

This paper investigates the interplay between numerical semigroups and enumerative combinatorics through the lens of Young diagrams, rook polynomials, and the Lah numbers. For a given numerical semigroup S, we associate a Young diagram constructed from the gap set of S. We then compute the rook polynomial corresponding to this diagram and analyze its coefficients. It is observed that these coefficients exhibit a strong connection with the Lah numbers, which count ordered partitions. Our approach introduces a new combinatorial interpretation of numerical semigroup gaps and reveals novel structural links between algebraic and enumerative concepts.