J. M. Marin, J. Ramirez Alfonsin, M. P. Revuelta
In this paper we compute the Frobenius number of certain {\em Fibonacci numerical semigroups}, that is, numerical semigroups generated by a set of Fibonacci numbers, in terms of Fibonacci numbers.
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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 $n_g$ tends to $S ϕ^g$, where $ϕ$ 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 $g \rightarrow \infty$, where $m$ is the multiplicity and $f$ is the Frobenius number.
The extended Frobenius problem for Fibonacci sequences incremented by a Fibonacci number
本文研究由斐波那契数列平移一个斐波那契数所得序列的扩展 Frobenius 问题，并证明相应数值半群族满足 Wilf 猜想。
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. The size of its complement is called the genus and the largest number in the complement is called its Frobenius number. We consider the set of numerical semigroups with a fixed Frobenius number $f$ and analyse their genus. We find the asymptotic distribution of genus in this set of numerical semigroups and show that it is a product of a Gaussian and a power series. We show that almost all numerical semigroups with Frobenius number $f$ have genus close to $\frac{3f}{4}$. We denote the number of numeric…
Numerical semigroups from rational matrices IV: computation of the matricial dimensions of numerical semigroups with small Frobenius number or genus
本文提出模论与线性规划方法，计算了所有 Frobenius 数≤10或亏格≤6的数值半群的矩阵维数，突破了以往技术局限。
The Frobenius problem for a class of quotients of numerical semigroups
本文研究了数值半群商的阿佩尔集特征，推导了半开形式的弗罗贝尼乌斯数和亏格公式，并在特定参数下得到某些商的弗罗贝尼乌斯数显式公式。
A generalised numerical semigroup (GNS) is a submonoid $S$ of $\mathbb{N}^d$ for which the complement $\mathbb{N}^d\setminus S$ is finite. The points in the complement $\mathbb{N}^d\setminus S$ are called gaps. A gap $F$ is considered Frobenius allowable if there is some relaxed monomial ordering on $\mathbb{N}^d$ with respect to which $F$ is the largest gap. We characterise the Frobenius allowable gaps of a GNS. A GNS that has only one Frobenius allowable gap is called a Frobenius GNS. We estimate the number of Frobenius GNS with a given Frobenius gap $F=(F^{(1)},\dots,F^{(d)})\in\mathbb{N}^d…
In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even i…
This paper proposes a new, visual method to study numerical semigroups and the Frobenius problem. The method is based on building a so-called reduction graph, whose nodes usually correspond to monogenic semigroups, and whose edges can have multiple inputs and outputs. If such a construction is possible, then determining whether the studied semigroup is symmetric, or finding explicit forms of its Apéry set and Hilbert series, is reduced to straightforward computations assisted by a MAPLE program we made available on arXiv. This approach applies to many of the cases considered in literature, inc…
Sub-Fibonacci behavior in numerical semigroup enumeration
本文利用 Kunz 坐标和赋权图同态结果，证明了数值半群数量偏离完美指数增长因子的指数上下界，获得了已知最精确的渐近界并揭示了次 Fibonacci 行为。
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical semigroups have genus equal to $g$? We outline Zhai's proof of a conjecture of Bras-Amorós that this sequence has Fibonacci-like growth. We now know that this sequence asymptotically grows as fast as the Fibonacci numbers, but it is still not known whether it is nondecreasing. We discuss this and other open problems. We highlight the many contributions made by undergraduates to problems in this area.
Counting numerical semigroups by Frobenius number, multiplicity, and depth
本文推广Backelin结果，给出固定Frobenius数和深度或重数的数值半群计数渐近公式，并解决相关猜想。
The pseudo-Frobenius numbers of a numerical semigroup are those gaps of the numerical semigroup that are maximal for the partial order induced by the semigroup. We present a procedure to detect if a given set of integers is the set of pseudo-Frobenius numbers of a numerical semigroup and, if so, to compute the set of all numerical semigroups having this set as set of pseudo-Frobenius numbers.
The problem of invariance and self-similarity in Z-modules is investigated. For a selection of examples relevant to quasicrystals, especially Fibonacci modules, we determine the semigroup of self-similarities and encapsulate the number of similarity submodules in terms of Dirichlet series generating functions.
We study how certain invariants of numerical semigroups relate to the number of second kind gaps. Furthermore, given two fixed non-negative integers F and k, we provide an algorithm to compute all the numerical semigroups whose Frobenius number is F and which have exactly k second kind gaps.
Delorme suggested that the set of all complete intersection numerical semigroups can be computed recursively. We have implemented this algorithm, and particularized it to several subfamilies of this class of numerical semigroups: free and telescopic numerical semigroups, and numerical semigroups associated to an irreducible plane curve singularity. The recursive nature of this procedure allows us to give bounds for the embedding dimension and for the minimal generators of a semigroup in any of these families.
Let $\mathbb{N}^{d}$ be the $d$-dimensional monoid of non-negative integers. A generalized numerical semigroup is a submonoid $ S\subseteq \mathbb{N}^d$ such that $H(S)=\mathbb{N}^d \setminus S$ is a finite set. We introduce irreducible generalized numerical semigroups and characterize them in terms of the cardinality of a special subset of $H(S)$. In particular, we describe relaxed monomial orders on $\mathbb N^d$, define the Frobenius element of $S$ with respect to a given relaxed monomial order, and show that the Frobenius element of $S$ is independent of the order if the generalized numeri…
$p$-numerical semigroup of generalized Fibonacci triples
本文针对非负整数 $p$，给出了涉及 Jacobsthal 多项式（包含 Fibonacci 数）的广义 Fibonacci 数值半群的 $p$-Frobenius 数和 $p$-亏格显式公式，并探讨了其在 $p$-Hilbert 级数上的应用。
In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating 0-1 vectors with as many component as the Frobenius number.
The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindström factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutat…
Frobenius problem and its many generalizations have been extensively studied in several areas of mathematics. We study semigroups of totally positive algebraic integers in totally real number fields, defining analogues of the Frobenius numbers in this context. We use a geometric framework recently introduced by Aliev, De Loera and Louveaux to produce upper bounds on these Frobenius numbers in terms of a certain height function. We discuss some properties of this function, relating it to absolute Weil height and obtaining a lower bound in the spirit of Lehmer's conjecture for algebraic vectors …
A numerical set is a co-finite subset of the natural numbers that contains zero. Its Frobenius number is the largest number in its complement. Each numerical set has an associated semigroup $A(T)=\{t\mid t+T\subseteq T\}$, which has the same Frobenius number as $T$. For a fixed Frobenius number $f$ there are $2^{f-1}$ numerical sets. It is known that there is a number $γ$ close to $0.484$ such that the ratio of these numerical sets that are mapped to $N_f=\{0\}\cup(f,\infty)$ is asymptotically $γ$. We identify a collection of families $N(D,f)$ of numerical semigroups such that for a fixed $D$ …
Internal numerical semigroups
本文研究数值半群的树结构，提出构造固定重数、弗罗贝尼乌斯数或亏格的内部半群的算法，猜想内部半群数量多于叶半群，还给出同时固定重数与弗罗贝尼乌斯数时两类半群数量的部分闭式公式。
This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.
On the Frobenius Number and Genus of a Collection of Semigroups Generalizing Repunit Numerical Semigroups
本文研究了特定数数半群的弗罗贝尼乌斯数和亏格，提出d可为负整数的新方法，并针对特殊情形如梅森、塔比特和雷努特数数半群给出简化公式，部分解决了普罗思数数半群的开放问题。
We present a procedure to enumerate the whole set of numerical semigroups with a given Frobenius number F, S(F). The methodology is based on the construction of a partition of S(F) by a congruence relation. We identify exactly one irreducible and one homogeneous numerical semigroup at each class in the relation, and from those two elements we reconstruct the whole class. An alternative more efficient method is proposed based on the use of the Kunz-coordinates vectors of the elements in S(F).
We examine properties of random numerical semigroups under a probabilistic model inspired by the Erdos-Renyi model for random graphs. We provide a threshold function for cofiniteness, and bound the expected embedding dimension, genus, and Frobenius number of random semigroups. Our results follow, surprisingly, from the construction of a very natural shellable simplicial complex whose facets are in bijection with irreducible numerical semigroups of a fixed Frobenius number and whose $h$-vector determines the probability that a particular element lies in the semigroup.
Affine semigroups without consecutive small elements
本文推广无连续小元素的数值半群概念至仿射锥的有限补子幺半群，给出计算给定Frobenius元素及重数的所有此类半群的算法，并分析生成元系统及最大嵌入维数情形下的分类。
In this work we introduce the notion of almost-symmetry for generalized numerical semigroups. In addition to the main properties occurring in this new class, we present several characterizations for its elements. In particular we show that this class yields a new family of Frobenius generalized numerical semigroups and extends the class of irreducible generalized numerical semigroups. This investigation allows us to provide a method of computing all almost symmetric generalized numerical semigroup having a fixed Frobenius element and organizing them in a rooted tree depending on a chosen monom…
$p$-numerical semigroups with $p$-symmetric properties
本文引入$p$-数值半群，推广数值半群理论，定义$p$-间隙、$p$-对称半群等，并研究其性质，当$p=0$时还原为经典情形。
In this paper, we bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to $m$-extensions. It allows us to identify gapsets and, in general, $m$-extensions with tilings of boards. As a consequence, we prove a version of Bras-Amorós conjecture for $m$-extensions. Besides, we obtain a lower bound for the number of gapsets with fixed genus and depth at most 3 and a family of upper bounds for the number of gapsets with fixed genus. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multipli…
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 catenary degree is an invariant that measures the distance between factorizations of elements within a numerical semigroup. In general, all possible catenary degrees of the elements of the numerical semigroups occur as the catenary degree of one of its Betti elements. In this study, Betti elements of some telescopic numerical semigroup families with embedding dimension three were found and formulated. Then, with the help of these formulas, Frobenius numbers and genus of these families were obtained. Also, the catenary degrees of telescopic numerical semigroups were found with the help of f…
A numerical semigroup is a sub-semigroup of the natural numbers that has a finite complement. Some of the key properties of a numerical semigroup are its Frobenius number F, genus g and type t. It is known that for any numerical semigroup $\frac{g}{F+1-g}\leq t\leq 2g-F$. Numerical semigroups with $t=2g-F$ are called almost symmetric, we introduce a new property that characterises them. We give an explicit characterisation of numerical semigroups with $t=\frac{g}{F+1-g}$. We show that for a fixed $α$ the number of numerical semigroups with Frobenius number $F$ and type $F-α$ is eventually cons…
Combinatorics on multisets is used to deduce new upper and lower bounds on the number of numerical semigroups of each given genus, significantly improving existing ones. In particular, it is proved that the number $n_g$ of numerical semigroups of genus $g$ satisfies $2F_{g}\leq n_g\leq 1+3\cdot 2^{g-3}$, where $F_g$ denotes the $g$th Fibonacci number.
On atoms of the set of generalized numerical semigroups with fixed corner element
本文研究原子广义数值半群，引入角特殊间隙概念，利用基数与极大性质刻画该类，并给出Frobenius广义数值半群不可约性及ANI的充分条件和充要条件。
Multiparameter counting of numerical semigroups: recurrences and leaf-discriminating trees
研究数值半群多参数计数问题，证明特定条件下的计数公式，提出种子算法的两种改编用于探索相关树结构，并改进算法策略，扩展对序列的认识，计算了多种情况下的半群数量及多参数分解。
Let n_g denote the number of numerical semigroups of genus g. Bras-Amoros 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.
Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset $\mbox{ Sg}_{\geq k}(A)$ of all vectors $b \in \mbox{ Sg}(A)$ such that $P_A(b) \cap {\mathbb Z}^n $ has at least $k$ solutions. …
A numerical semigroup is an additive subsemigroup of the non-negative integers. In this paper, we consider parametrized families of numerical semigroups of the form $P_n = \langle f_1(n), \ldots, f_k(n) \rangle$ for polynomial functions $f_i$. We conjecture that for large $n$, the Betti numbers, Frobenius number, genus, and type of $P_n$ each coincide with a quasipolynomial. This conjecture has already been proven in general for Frobenius numbers, and for the remaining quantities in the special case when $P_n = \langle n, n + r_2, \ldots, n + r_k \rangle$. Our main result is to prove our conje…
On counting numerical semigroups by maximum primitive and Wilf's conjecture
本文提出通过最大原始数计数数值半群的方法，并证明其与通过Frobenius数计数的关系。同时证明大尺寸最大原始数的数值半群几乎都满足Wilf猜想。
Our goal is to convince the readers that the theory of complex normal surface singularities can be a powerful tool in the study of numerical semigroups, and, in the same time, a very rich source of interesting affine and numerical semigroups. More precisely, we prove that the strongly flat semigroups, which satisfy the maximality property with respect to the Diophantine Frobenius problem, are exactly the numerical semigroups associated with negative definite Seifert homology spheres via the possible 'weights' of the generic $S^1$-orbit. Furthermore, we consider their generalization to the Seif…
The Frobenius Problem for the Proth Numbers
本文定义了Proth数值半群$P_{k}(n)$，给出了其嵌入维数公式，并通过推导Frobenius数的闭式公式解决了该半群的Frobenius问题，同时证明了其满足Wilf性质。
In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of t…
We present procedures to calculate the set of Arf numerical semigroups with given genus, given conductor and given genus and conductor. We characterize the Kunz coordinates of an Arf numerical semigroup. We also describe Arf numerical semigroups with fixed Frobenius number and multiplicity up to six.
A Frobenius problem suggested by prime $k$-tuplets (extended version)
本文研究由素数k元组（尤其是素数三元组和四元组）提出的Frobenius问题，并分析相关数值半群的性质。
We improve the previously best known lower and upper bounds on the number n_g of numerical semigroups of genus g. Starting from a known recursive description of the tree T of numerical semigroups, we analyze some of its properties and use them to construct approximations of T by generating trees whose nodes are labeled by certain parameters of the semigroups. We then translate the succession rules of these trees into functional equations for the generating functions that enumerate their nodes, and solve these equations to obtain the bounds. Some of our bounds involve the Fibonacci numbers, and…
The shape of a random numerical semigroup
研究随机数值半群的统计性质，分析其图形结构，证明当 genus 趋向无穷时，点集接近两条线段的联合。
Let 0 < a < b be two relatively prime integers and let <a,b> be the numerical semigroup generated by a and b with Frobenius number g(a,b)=ab-a-b. In this note, we prove that there exists a prime number p in <a,b> with p < g(a,b) when the product ab is sufficiently large. Two related conjectures are posed and discussed as well.
For a numerical semigroup, we encode the set of primitive elements that are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an efficient algorithm for exploring the tree up to a given genus. The algorithm exploits the second nonzero element of a numerical semigroup and the particular pseudo-ordinary case in which this element is the conductor.
We find a relation between the genus of a quotient of a numerical semigroup $S$ and the genus of $S$ itself. We use this identity to compute the genus of a quotient of $S$ when $S$ has embedding dimension $2$. We also exhibit identities relating the Frobenius numbers and the genus of quotients of numerical semigroups that are generated by certain types of arithmetic progressions.
A Note on Generalized Repunit Numerical Semigroups
本文研究广义循环单位数值半群的Frobenius问题，在特定条件下给出Frobenius数和亏格的公式，并简化了循环单位、梅森和塔比特等特殊情形。
Numerical semigroups are cofinite additive submonoids of the natural numbers. In 2011, Keith and Nath illustrated an injection from numerical semigroups to integer partitions. We explore this connection between partitions and numerical semigroups with a focus on classifying the partitions that appear in the image of the injection from numerical semigroups. In particular, we count the number of partitions that correspond to numerical semigroups in terms of genus, Frobenius number, and multiplicity, with some restrictions.
On the Frobenius Number of Quotients of Numerical Semigroups
研究数值半群商的弗罗贝尼乌斯数问题，通过受柯蒂斯定理启发，利用狄利克雷定理，得到该数不能用有限多项式公式统一表示、特定函数性质及不存在相关非零多项式等结果。
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 in the translation of a numerical semigroup to its so-called Kunz-coordinates vector.
Steganographic information hiding via symmetric numerical semigroups
本文提出一种基于对称数值半群的隐写信息隐藏方案，利用间隙结构实现信息嵌入，通过数论方法实现隐蔽通信和抗量子安全信息隐藏。
Let $f_1(n), \ldots, f_k(n)$ be polynomial functions of $n$. For fixed $n\in\mathbb{N}$, let $S_n\subseteq \mathbb{N}$ be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)$. As $n$ varies, we show that many invariants of $S_n$ are eventually quasi-polynomial in $n$, such as the Frobenius number, the type, the genus, and the size of the $Δ$-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups $S_n\subseteq \mathbb{N}^m$ generated be vectors whose coo…
The $p$-numerical semigroup of the triple of arithmetic progressions
本文研究算术数列三元组的 $p$-Frobenius 数，通过确定 $p$-Apéry 集的元素，给出了此前未知的 $p>0$ 情形下显式公式及相关值。
The Frobenius number for shifted geometric sequences associated with the number of solutions
本文给出了移位几何序列$\{a b^n-c\}_n$的$p$-Frobenius数的闭式公式，涵盖几何、Thabit和Mersenne序列及其变体，解决了$p>0$时广义Frobenius数公式缺失的问题。
$p$-numerical semigroups of Pell triples
本文针对基于 Pell 型数递推关系构建的 Pell 数值半群，利用 $p$-Apéry 集，通过逐案讨论推导出了其 $p$-Frobenius 数和 $p$-亏格的显式公式。
Given coprime positive integers $g_1 < \ldots < g_e$, the Frobenius number $F=F(g_1,\ldots,g_e)$ is the largest integer not representable as a linear combination of $g_1,\ldots,g_e$ with non-negative integer coefficients. Let $n$ denote the number of all representable non-negative integers less than $F$; Wilf conjectured that $F+1 \le e n$. We provide bounds for $g_1$ and for the type of the numerical semigroup $S=\langle g_1,\ldots,g_e \rangle$ in function of $e$ and $n$, and use these bounds to prove that $F+1 \le q e n$, where $q= \left \lceil \frac{F+1}{g_1} \right \rceil$, and $F+1 \le e …
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