Counting numerical semigroups by genus and even gaps via Kunz-coordinate vectors

TL;DR

This paper establishes a bijection between certain numerical semigroups and points in a rational polytope, providing a new approach to analyze the sequence counting semigroups by genus.

Contribution

It introduces a novel correspondence between numerical semigroups with specific properties and rational polytope points, aiding in understanding their enumeration.

Findings

Provides a method to study the sequence of semigroups by genus

Offers a geometric perspective via polytope correspondence

Aims to determine if the sequence of counts is increasing

Abstract

We contruct a one-to-one correspondence between a subset of numerical semigroups with genus $g$ and $\gamma$ even gaps and the integer points of a rational polytope. In particular, we give an overview to apply this correspondence to try to decide if the sequence $(n_g)$ is increasing, where $n_g$ denotes the number of numerical semigroups with genus $g$.

Equations

$${\mathbf{x}}_g:{\mathcal{S}}_{\gamma }(g)\to {\mathcal{S}}_{\gamma },S\mapsto S/2,$$

$${\mathbf{x}}_{g_d}:{\mathcal{S}}_{(d,\gamma )}(g)\to {\mathcal{S}}_{\gamma },S\mapsto S/d,$$

$$\text{Is it true that }{n}_g\le n_{g+1},\text{ for all }g?$$

$$n_g=\sum _{\gamma =0}^{\lfloor 2g/3\rfloor }{N}_{\gamma }(g).$$

$$S=⟨m,w_1,w_2,\dots ,w_{m-1}⟩.$$

$$\{\begin{array}{l}(3e_1+1)+(3e_1+1)\ge 3e_2+2\\ (3e_2+2)+(3e_2+2)\ge 3e_1+1.\end{array}$$

$$\{\begin{array}{l}2X_1\ge X_2\\ 2X_2+1\ge X_1\\ X_1+X_2=g.\end{array}$$

$$\{\begin{array}{l}{X}_i+X_j\ge X_{i+j},\text{ for }1\le i\le j\le m-1;i+j<m;\\ X_i+X_j+1\ge X_{i+j-m},\text{ for }1\le i\le j\le m-1;i+j>m\\ {\sum }_{k=1}^{m-1}{X}_k=g.\end{array}$$

$$N_{\gamma }(g)=\sum _{T\in {\mathcal{S}}_{\gamma }}\#{\mathbf{x}}_g^{-1}(T).$$

$${\mathcal{S}}_{\gamma }=\bigcup _{m=2}^{\gamma +1}{\mathcal{S}}_{\gamma }^m.$$

$$N_{\gamma }(g)=\sum _{m=2}^{\gamma +1}\sum _{T\in {\mathcal{S}}_{\gamma }^m}\#{\mathbf{x}}_g^{-1}(T).$$

$$T=⟨m,m{e}_1+1,m{e}_2+2,\dots ,m{e}_{m-1}+(m-1)⟩,$$

$$S=⟨2m,2m{e}_1+2,\dots ,2m{e}_{m-1}+(2m-2),2m{k}_1+1,2m{k}_3+3,\dots ,2m{k}_{2m-1}+(2m-1)⟩,$$

$$\{\begin{array}{l}(\ast )\{\begin{array}{l}{X}_{2i-1}+e_j\ge X_{2(i+j)-1},\text{ for }1\le i\le m;1\le j\le m-1;i+j\le m;\\ X_{2i-1}+e_j+1\ge X_{2(i+j-m)-1},\text{ for }1\le i\le m;1\le j\le m-1;i+j>m;\end{array}\\ (\ast \ast )\{\begin{array}{l}{X}_{2i-1}+X_{2j-1}\ge e_{i+j-1},\text{ for }1\le i\le j\le m;i+j\le m;\\ X_{2i-1}+X_{2j-1}+1\ge e_{i+j-1-m},\text{ for }1\le i\le j\le m;i+j\ge m+2;\end{array}\\ {\sum }_{i=1}^m{X}_{2i-1}=g-\gamma ,\end{array}$$

$$S=⟨4,4\gamma +2,4k_1+1,4k_3+3⟩,$$

$$(\#)\{\begin{array}{l}(\ast )\{\begin{array}{l}-\gamma -1\le X_3-X_1\le \gamma \end{array}\\ (\ast \ast )\{\begin{array}{l}{X}_1+X_1\ge \gamma \\ X_3+X_3+1\ge \gamma \end{array}\\ X_1+X_3=g-\gamma .\end{array}$$

$$N_{\gamma }^2(g)=\{\begin{array}{l}0,\text{ if }g<2\gamma \\ k+1,\text{ if }g=2\gamma +k\text{ and }k\in \{0,1,\dots ,\gamma -1\}\\ \gamma +1,\text{ if }g\ge 3\gamma .\end{array}$$

$$\text{Let }\gamma \in \mathbb{N}\text{ and }m\in [2,\gamma +1]\cap \mathbb{Z}.\text{ Is it true that }{N}_{\gamma }^m(g)\le N_{\gamma }^m(g+1),\text{ for all }g?$$

Full text

2010 Math. Subj. Class.: Primary 20M14; Secondary 05A15, 05A19

Keywords: numerical semigroup, multiplicity, even gap, genus, Apéry set, Kunz-coordinate vector

1. Introduction

A numerical semigroup $S$ is a subset of $\mathbb{N}_{0}$ such that $0\in S$, it is closed under addition and the set $G(S):=\mathbb{N}_{0}\setminus S$, the set of gaps of $S$, is finite. The number of elements $g=g(S)$ of $G(S)$ is called the genus of $S$ and the first non-zero element in $S$ is called the multiplicity of $S$. If $S$ is a numerical semigroup with genus $g$ then one can ensure that all gaps of $S$ belongs to $[1,2g]$; in particular, $\{2g+i:i\in\mathbb{N}_{0}\}\subseteq S$ and the number of numerical semigroups with genus $g$, denoted by $n_{g}$, is finite. Some excellent references for the background on numerical semigroups are the books [5] and [7].

Troughtout this paper, we keep the notation proposed by Bernardini and Torres [1]: the set of numerical semigroups with genus $g$ is denoted by $\mathcal{S}_{g}$ and has $n_{g}$ elements and the the set of numerical semigroups with genus $g$ and $\gamma$ even gaps is denoted by $\mathcal{S}_{\gamma}(g)$ and has $N_{\gamma}(g)$ elements.

In this paper we use the quite useful parametrization

[TABLE]

where $S/2:=\{s\in\mathbb{N}_{0}:2s\in S\}$.

Naturally, the set $\mathcal{S}_{\gamma}(g)$ and the map $\mathbf{x}_{g}$ can be generalized. Let $d>1$ be an integer. The set of numerical semigroups with genus $g$ and $\gamma$ gaps which are congruent to 0 modulo $d$ is denoted by $\mathcal{S}_{(d,\gamma)}(g)$. There is a natural parametrization given by

[TABLE]

where $S/d:=\{s\in\mathbb{N}_{0}:ds\in S\}$. This concept appears in [8], for instance.

In this paper, we obtain a one-to-one correspondence between the set $\mathbf{x}_{g}^{-1}(T)$ and the integer points of a rational polytope.

As an application of this correspondence, we give a new approach to compute the numbers $N_{\gamma}(g)$. Our main goal is finding a new direction to discuss the following question.

[TABLE]

The first few elements of the sequence $(n_{g})$ are $1,1,2,4,7,12,23,39,67$. Kaplan [6] wrote a nice survey on this problem and one can find information of these numbers in Sloane’s On-line Encyclopedia of Integer Sequences [10].

Bras-Amorós [3] conjectured remarkable properties on the behaviour of the sequence $(n_{g})$:

1. (1)

$\lim_{g\to\infty}\frac{n_{g+1}+n_{g}}{n_{g+2}}=1$; 2. (2)

$\lim_{g\to\infty}\frac{n_{g+1}}{n_{g}}=\varphi:=\frac{1+\sqrt{5}}{2}$; 3. (3)

$n_{g+2}\geq n_{g+1}+n_{g}$ for any $g$.

Zhai [12] proved that $\lim_{g\to\infty}n_{g}\varphi^{-g}$ is a constant. As a consequence, it confirms that items (1) and (2) hold true. However, item (3) is still an open problem; even a weaker version, proposed at (1.2), is an open question. Zhai’s result also ensures that $n_{g}<n_{g+1}$ for large enough $g$. Fromentin and Hivert [4] verified that $n_{g}<n_{g+1}$ also holds true for $g\leq 67$.

Torres [11] proved that $\mathcal{S}_{\gamma}(g)\neq\emptyset$ if, and only if, $2g\geq 3\gamma$. Hence,

[TABLE]

In order to work on Question (1.2), Bernardini and Torres [1] tried to understand the effect of the even gaps on a numerical semigroup. By using the so-called $t$-translation, they proved that $N_{\gamma}(g)=N_{\gamma}(3\gamma)$ for $g\geq 3\gamma$ and also $N_{\gamma}(g)<N_{\gamma}(3\gamma)$ for $g<3\gamma$. Although numerical evidence points out that $N_{\gamma}(g)\leq N_{\gamma}(g+1)$ holds true for all $g$ and $\gamma$, their methods could not compare numbers $N_{\gamma}(g_{1})$ and $N_{\gamma}(g_{2})$, with $3\gamma/2\leq g_{1}<g_{2}<3\gamma$. Notice that if $N_{\gamma}(g_{1})\leq N_{\gamma}(g_{2})$, for $3\gamma/2\leq g_{1}<g_{2}<3\gamma$ then $n_{g}<n_{g+1}$, for all $g$.

2. Apéry set and Kunz-coordinate vector

Let $S$ be a numerical semigroup and $n\in S$. The Apéry set of $S$ (with respect to $n$) is the set $Ap(S,n)=\{s\in S:s-n\notin S\}$. If $n=1$, then $S=\mathbb{N}_{0}$ and $Ap(\mathbb{N}_{0},1)=\{0\}$. If $n>1$, then a there are $w_{1},\ldots,w_{n-1}\in\mathbb{N}$ such that $Ap(S,n)=\{0,w_{1},\ldots,w_{n-1}\}$, where $w_{i}=\min\{s\in S:s\equiv i\pmod{n}\}$.

Proposition 2.1**.**

Let $S$ be a numerical semigroup with multiplicity $m$ and $Ap(S,m)=\{0,w_{1},\ldots,w_{m-1}\}$. Then

[TABLE]

Proof.

It is clear that $am\in\langle m,w_{1},w_{2},\ldots,w_{m-1}\rangle,\forall a\in\mathbb{N}.$ For $s\in S$, $m\nmid s$, there is $\tilde{k}\in\mathbb{N}_{0}$ such that $s=w_{i}+\tilde{k}m\in\langle m,w_{1},w_{2},\ldots,w_{m-1}\rangle$. On the other hand, $m,w_{1},\ldots,w_{m-1}\in S$. ∎

Let $S$ be a numerical semigroup, $n\in S$ and consider $Ap(S,n)=\{0,w_{1},\ldots,w_{n-1}\}$. There are $e_{1},\ldots,e_{n-1}\in\mathbb{N}$ such that $w_{i}=ne_{i}+i$, for each $i\in\{1,\ldots,n-1\}$. The vector $(e_{1},\ldots,e_{n-1})\in\mathbb{N}_{0}^{n-1}$ is called the Kunz-coordinate vector of $S$ (with respect to $n$). In particular, if $m$ is the multiplicity of $S$, then the Kunz-coordinate vector of $S$ (with respect to $m$) is in $\mathbb{N}^{m-1}$. This concept appears in [2], for instance.

A natural task is finding conditions for a vector $(x_{1},\ldots,x_{m-1})\in\mathbb{N}^{m-1}$ to be a Kunz-coordinate vector (with respect to the multiplicity $m$ of $S$) of some numerical semigroup $S$ with multiplicity $m$. The following examples illustrate the general method, which is presented in Proposition 2.4.

Example 2.2**.**

Numerical semigroups with multiplicity $2$ are $\langle 2,2e_{1}+1\rangle$, where $e_{1}\in\mathbb{N}$.

There is a one-to-one correspondence between the set of numerical semigroups with multiplicity $2$ and the set of positive integers given by $\langle 2,2e_{1}+1\rangle\mapsto e_{1}$.

Example 2.3**.**

Let $S=\langle 3,3e_{1}+1,3e_{2}+2\rangle$ be a numerical semigroup with multiplicity $3$ and genus $g$, where $e_{1},e_{2}\in\mathbb{N}$. By minimality of $w_{1}=3e_{1}+1$ and $w_{2}=3e_{2}+2$, $(e_{1},e_{2})$ satisfies

[TABLE]

The set of gaps of $S$ has $e_{1}+e_{2}$ elements, since $G(S)=\{3n_{1}+1:0\leq n_{1}<e_{1}\}\cup\{3n_{2}+2:0\leq n_{2}<e_{2}\}$. Thus, $e_{1}+e_{2}=g$. On the other hand, if $(e_{1},e_{2})\in\mathbb{N}^{2}$ is such that $2e_{1}\geq e_{2}$, $2e_{2}+1\geq e_{1}$ and $e_{1}+e_{2}=g$, then $\langle 3,3e_{1}+1,3e_{2}+2\rangle$ is a numerical semigroup with multiplicity $m$ and genus $g$.

Hence, there is a one-to-one correspondence between the set of numerical semigroups with multiplicity $3$ and the vectors of $\mathbb{N}^{2}$ which are solutions of

[TABLE]

In order to give a characterization of numerical semigroups with fixed multiplicity and fixed genus, the main idea is generalizing Example 2.3. The following is a result due to Rosales et al. [8].

Proposition 2.4**.**

There is a one-to-one correspondence between the set of numerical semigroups with multiplicity $m$ and genus $g$ and the positive integer solutions of the system of inequalities

[TABLE]

Let $S=\langle m,w_{1},\ldots,w_{m-1}\rangle$ be a numerical semigroup with multiplicity $m$ and genus $g$, where $w_{i}=me_{i}+i$. The main idea of the proof is using the minimality of $w_{1},\ldots,w_{m-1}$ and observing that $w_{i}+w_{j}\equiv i+j\pmod{m}$ and $G(S)=\bigcup_{i=1}^{m-1}\{mn_{i}+i:0\leq n_{i}<e_{i}\}$. For a full proof, see [8].

3. The main result and an application to a counting problem

In [1], the calculation of $N_{\gamma}(g)$ was given by

[TABLE]

In this section, we present a new way for computing those numbers. In order to do this, we fix the multiplicity of $T\in\mathcal{S}_{\gamma}$.

First of all, we obtain a relation between the genus and the multiplicity of a numerical semigroup.

Proposition 3.1**.**

Let $S$ be a numerical semigroup with genus $g$ and multiplicity $m$. Then $m\leq g+1$.

Proof If a numerical semigroup $S$ has multiplicity $m$ and genus $g$ with $m\geq g+2$, then the number of gaps of $S$ would be, at least, $g+1$ and it is a contradiction. Hence $m\leq g+1$.

Remark 3.2*.*

The bound obtained in Proposition 3.1 is sharp, since $\{0,g+1,\ldots\}$ has genus $g$ has multiplicity $g+1$.

If $\gamma=0$, then $\mathcal{S}_{0}=\{\mathbb{N}_{0}\}$ and $\mathbf{x}_{g}^{-1}(\mathbb{N}_{0})=\{\langle 2,2g+1\rangle\}$. Hence, $N_{0}(g)=1$, for all $g$. If $\gamma>0$, we divide the set $\mathcal{S}_{\gamma}$ into the subsets $\mathcal{S}_{\gamma}^{m}:=\{S:g(S)=\gamma\text{ and }m(S)=m\}$, where $m\in[2,\gamma+1]\cap\mathbb{Z}$. We can write

[TABLE]

Putting (3.1) and (3.2) together, we obtain

[TABLE]

Thus, it is important to give a characterization for $T\in\mathcal{S}_{\gamma}^{m}$. We can describe $T$ by its Apéry set (with respect to its multiplicity $m$) and write

[TABLE]

where $me_{i}+i=\min\{s\in S:s\equiv i\pmod{m}\}$.

The next result characterizes all numerical semigroups of $\mathbf{x}_{g}^{-1}(T)$, for $T\in\mathcal{S}_{\gamma}^{m}$. It is a consequence of Proposition 2.4.

Theorem 3.3**.**

Let $T=\langle m,me_{1}+1,\ldots,me_{m-1}+(m-1)\rangle\in\mathcal{S}_{\gamma}^{m}$. A numerical semigroup $S$ belongs to $\mathbf{x}_{g}^{-1}(T)$ if, and only if,

[TABLE]

where $(k_{1},k_{3},\ldots,k_{2m-1})\in\mathbb{N}_{0}^{m}$ satisfies the system

[TABLE]

Proof The even numbers $2m,2me_{1}+2,\ldots,2me_{m-1}+2me_{m-1}+2(m-1)$ belongs to $Ap(2m,S)$. Let $2mk_{1}+1,2mk_{3}+3,\ldots,2mk_{2m-1}+(2m-1)$ be the odd numbers of $Ap(2m,S)$. Thus, $(e_{1},k_{1},e_{2},k_{3},\ldots,e_{m-1},k_{2m-1})\in\mathbb{N}_{0}^{m-1}$ is the Kunz-coordinate vector of $S$ (with respect to $2m$).

Now, we apply Proposition 2.4. Inequalites given in $(*)$ come from sums of an odd element of $Ap(2m,S)$ with an even element of $Ap(2m,S)$, while inequalities given in $(**)$ come from sums of two odd elements of $Ap(2m,S)$. Since $(e_{1},\ldots,e_{m-1})$ is the Kunz-coordinate vector of $T$ (with respect to $m$), then the sum of two even elements of $Ap(2m,S)$ belongs to $S$. Finally, last equality comes from the fact that $S$ has $g-\gamma$ odd gaps.

Remark 3.4*.*

Some of the numbers $k_{i}$ can be zero. Hence, it is possible that the multiplicity of $S$ is not $2m$.

Example 3 Let $T=\langle 2,2\gamma+1\rangle\in\mathcal{S}_{\gamma}^{2}$, with $\gamma\in\mathbb{N}$. Theorem 3.3 ensures that if $S\in\mathbf{x}_{g}^{-1}(T)$, then

[TABLE]

where $(k_{1},k_{3})\in\mathbb{N}_{0}^{2}$ satisties

[TABLE]

The set of integer points of this region is in one-to-one correspondence with the set $\{S\in\mathcal{S}_{\gamma}(g):S/2\text{ has multiplicity }2\}$.

If $g$ is fixed, then the set of points that satisfies the system $(\#)$ is a polytope (a line segment). We are interested in the set of integer points of this polytope. The following figure shows examples for some values of $g$. Each integer point represents a numerical semigroup of the set $\{S\in\mathcal{S}_{\gamma}(g):S/2\text{ has multiplicity }2\}$.

Let $N^{m}_{\gamma}(g)=\sum_{T\in\mathcal{S}_{\gamma}^{m}}\#\mathbf{x}_{g}^{-1}(T)$. After some computations, we obtain

[TABLE]

In particular, $N^{2}_{\gamma}(g)\leq N^{2}_{\gamma}(g+1)$. We leave the following open question.

[TABLE]

A positive answer to Question (3.3) implies a positive answer to Question (1.2).

Acknowledgment. The author was partially supported FAPDF-Brazil (grant 23072.91.49580.29052018). Part of this paper was presented in the “INdAM: International meeting on numerical semigroups” (2018) at Cortona, Italy. I am grateful to the referee for their comments, suggestions and corrections that allowed to improve this version of the paper.