On primality and atomicity of numerical power monoids

Abstract

In the first part of this paper, we establish a variation of a recent result by Bienvenu and Geroldinger on the (almost) non-existence of absolute irreducibles in (restricted) power monoids of numerical monoids: we argue the (almost) non-existence of primal elements in the same class of power monoids. The second part of this paper, devoted to the study of the atomic density of ${\mathcal{P}}_{\text{fin},0}({\mathbb{N}}_0)$, is motivated by work of Shitov, a recent paper by Bienvenu and Geroldinger, and some questions pointed out by Geroldinger and Tringali. In the same, we study atomic density through the lens of the natural partition $\{{\mathcal{A}}_{n,k}:k\in {\mathbb{N}}_0\}$ of ${\mathcal{A}}_n$, the set of atoms of ${\mathcal{P}}_{\text{fin},0}({\mathbb{N}}_0)$ with maximum at most $n$: \[ \mathcal{A}_{n,k} = \{A \in \mathcal{A} : \max A \le n \text{ and } |A| = k\} \] for all $n,k\in \mathbb{N}$, where $\mathcal{A}$ is the set of atoms of ${\mathcal{P}}_{\text{fin},0}({\mathbb{N}}_0)$. We pay special attention to the sequence $(\hat{I}{\pm }_{n,k}{)}_{n,k\ge 1}$, where $\hat{I}{\pm }_{n,k}$ denote the size of the block ${\mathcal{A}}_{n,k}$. First, we establish some bounds and provide some asymptotic results for $(\hat{I}{\pm }_{n,k}{)}_{n,k\ge 1}$. Then, we take some probabilistic approach to argue that, for each $n\in \mathbb{N}$, the sequence $(\hat{I}{\pm }_{n,k}{)}_{k\ge 1}$ is almost unimodal. Finally, for each $n\in \mathbb{N}$, we consider the random variable $X_n:{\mathcal{A}}_n\to {\mathbb{N}}_0$ defined by the assignments $X_n:A\mapsto |A|$, whose probability mass function is $\mathbb{P}(X_n=k)=\hat{I}{\pm }_{n,k}/|{\mathcal{A}}_n|$. We conclude proving that, for each $m\in \mathbb{N}$, the sequence of moments $(\mathbb{E}(X_n^m){)}_{n\ge 1}$ behaves asymptotically as that of a sequence $(\mathbb{E}(Y_n^m){)}_{n\ge 1}$, where $Y_n$ is a binomially distributed random variable with parameters $n$ and $\frac{1}{2}$.

17 pages