The set of numerical semigroups of a given genus

In this paper we present a new approach to construct the set of numerical semigroups with a fixed genus. Our methodology is based on the construction of the set of numerical semigroups with fixed Frobenius number and genus. An equivalence relation is given over this set and a tree structure is defined for each equivalence class. We also provide a more efficient algorithm based on the translation of a numerical semigroup to its so-called Kunz-coordinates vector.     

Constructing numerical semigroups of a given genus

Let n _g denote the number of numerical semigroups of genus g. Bras-Amorós conjectured that n _g possesses certain Fibonacci-like properties. Almost all previous attempts at proving this conjecture were based on analyzing the semigroup tree. We offer a new, simpler approach to counting numerical semigroups of a given genus. Our method gives direct constructions of families of numerical semigroups, without referring to the generators or the semigroup tree. In particular, we give an improved asymptotic lower bound for n _g .     

Fibonacci-like growth of numerical semigroups of a given genus

We give an asymptotic estimate of the number of numerical semigroups of a given genus. In particular, if n _g is the number of numerical semigroups of genus g, we prove that $$\lim_{g \rightarrow \infty} n_g \varphi^{-g} = S $$ where $\varphi = \frac{1 + \sqrt{5}}{2}$ is the golden ratio and S is a constant, resolving several related conjectures concerning the growth of n _g . In addition, we show that the proportion of numerical semigroups of genus g satisfying f<3m approaches 1 as g→∞, where m is the multiplicity and f is the Frobenius number.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

Fibonacci-like behavior of the number of numerical semigroups of a given genus

We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of semigroups of genus at most 50. The Wilf conjecture has also been checked for all numerical semigroups with genus in the same range.     

Counting numerical semigroups by genus and even gaps

Let $n_g$ be the number of numerical semigroups of genus $g$. We present an approach to compute $n_g$ by using even gaps, and the question: Is it true that $n_{g+1}>n_g$? is investigated. Let $N_{\gamma }\left(g\right)$ be the number of numerical semigroups of genus $g$ whose number of even gaps equals $\gamma$. We show that $N_{\gamma }\left(g\right)=N_{\gamma }\left(3\gamma \right)$ for $\gamma \le ?g∕3?$ and $N_{\gamma }\left(g\right)=0$ for $\gamma >?2g∕3?$; thus the question above is true provided that $N_{\gamma }(g+1)>N_{\gamma }\left(g\right)$ for $\gamma =?g∕3?+1,\dots ,?2g∕3?$. We also show that $N_{\gamma }\left(3\gamma \right)$ coincides with $f_{\gamma }$, the number introduced by Bras-Amorós (2012) in connection with semigroup-closed sets. Finally, the stronger possibility $f_{\gamma }～{\phi }^{2\gamma }$ arises being $\phi =(1+\sqrt{5})∕2$ the golden number.     

On the weight of numerical semigroups

We investigate the weights of a family of numerical semigroups by means of even gaps and the Weierstrass property for such a family.     

Atoms of the set of numerical semigroups with fixed Frobenius number

Given a positive integer g, we denote by F(g) the set of all numerical semigroups with Frobenius number g. The set (F(g),∩) is a semigroup. In this paper we study the generators of this semigroup.     

On a set of generating relations of the full transformation semigroups

A set of generating relations of the full transformation semigroup Σ_n consisting of all mappings of the set $\text{Ω = {1, …, n}}$ into itself with respect to a natural system of generators {γ_jⁱ, σ_jⁱ} (defined in Section 1) is given. The set of relations (1)～(15) given in Section 1 turns out to be a set of generating relations.     

On numerical semigroups and the order bound

Let S={s₀=0<s₁<?<s_i…}⊆N be a numerical non-ordinary semigroup; then set, for each . We find a non-negative integer m such that d_ORD(i)=ν_i+1 for i≥m, where d_ORD(i) denotes the order bound on the minimum distance of an algebraic geometry code associated to S. In several cases (including the acute ones, that have previously come up in the literature) we show that this integer m is the smallest one with the above property. Furthermore it is shown that every semigroup generated by an arithmetic sequence or generated by three elements is acute. For these semigroups, the value of m is also found.     

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

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

On the enumeration of circular maps with given number of edges

A map is a closed Riemann surface S with an embedded graph G such that S G is homeomorphic to a disjoint union of open disks. Tutte began a systematic study of maps in the 1960s, and contemporary authors are actively developing it. We introduce the concept of circular map and establish its equivalence to the concept of map admitting a coloring of the faces in two colors. The main result is a formula for the number of circular maps with given number of edges.     

On the least size of a graph with a given degree set

The degree set of a finite simple graph G is the set of distinct degrees of vertices of G. A theorem of Kapoor et al. [Degree sets for graphs, Fund. Math. 95 (1977) 189-194] asserts that the least order of a graph with a given degree set D is 1+max(D). We look at the analogous problem concerning the least size of a graph with a given degree set D. We determine the least size for the sets D when (i) |D|?3; (ii) D={1,2,…,n}; and (iii) every element in D is at least |D|. In addition, we give sharp upper and lower bounds in all cases.