Computation of numerical semigroups by means of seeds

Abstract

For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of seed by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus.

The right-generators descendant of a numerical semigroup

“…The RGD algorithm is based on a much simpler concept than the seeds algorithm in [6]. Although the pseudocode is not so short and concise, it is more suitable to implement in any programming language, since it does not require any bitwise operations through long integer data types.…”

“…In the next section we show how to compute efficiently from D(Λ) the RGD of the children of Λ. To this end, we restate here a special case of Lemma 2.1 in [6], where right generators correspond to order-zero seeds. The result complements Lemma 1.1 by dealing with new right generators, that is, with the right generators of a child that are not primitive elements of its parent.…”

Counting Numerical Semigroups

“…Fromentin and Hivert use a massively improved algorithm for computing the semigroup tree, utilizing depth first rather than breadth first search along with several specific technical optimizations, to compute N (g) for g ≤ 67. More recently, Bras-Amorós and Fernández-González have suggested a new algorithm based on seeds, which can be thought of as a generalization of the notion of strong and weak effective generators [11].…”

Counting numerical semigroups by genus and even gaps

“…Blanco and Rosales [1] approached this problem by considering a partition of S g by subsets of semigroups S of a given Frobenius number F = F (S), which by definition is the biggest integer which does not belong to S; see also [2]. In any case, computing the exact value of n g seems to be out of reach although there exist algorithmic methods for determining such a number [14], [3].…”