Related papers: On the smallest partition associated to a numerica…
A \emph{numerical semigroup} is a subset $\Lambda$ of the nonnegative integers that is closed under addition, contains $0$, and omits only finitely many nonnegative integers (called the \emph{gaps} of $\Lambda$). The collection of all…
The lattice diameter of a bounded set $S \subset \mathbb{R}^d$ measures the maximal number of lattice points in a segment whose endpoints are lattice points in $S$. Such a segment is called a lattice diameter segment of $S$. This simple…
Let $K\subseteq{\mathbb R}^n$ be a convex semialgebraic set. The semidefinite extension degree ${\mathrm{sxdeg}}(K)$ of $K$ is the smallest number $d$ such that $K$ is a linear image of an intersection of finitely many spectrahedra, each of…
Motivated by the many roles that hook lengths play in mathematics, we study the distribution of the number of $t$-hooks in the partitions of $n$. We prove that the limiting distribution is normal with mean $\mu_t(n)\sim…
Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regular…
There is a well-studied correspondence by Jaclyn Anderson between partitions that avoid hooks of length s or t and certain binary strings of length s+t. Using this map, we prove that the total size of a random partition of this kind…
Let $\lambda$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $\lambda$ to be the difference of its leg and arm lengths. Define $h_{1,1}(\lambda)$ to be the number of cells of $\lambda$ with hook…
A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer…
The paper is concerned with the asymptotic analysis of a family of Boltzmann (multiplicative) distributions over the set $\check{\varLambda}^{q}$ of strict integer partitions (i.e., with unequal parts) into perfect $q$-th powers. A…
Recently, Amdeberhan et al. proved congruences for the number of hooks of fixed even length among the set of self-conjugate partitions of an integer $n$, therefore answering positively a conjecture raised by Ballantine et al.. In this…
How to quantify the distance between any two partitions of a finite set is an important issue in statistical classification, whenever different clustering results need to be compared. Developing from the traditional Hamming distance between…
We explore the notion of sectional number of a group homomorphism, leading to a generalization of the covering number of a group, and present several characterizations when the sectional number is finite, providing estimates for computing…
We give relations between the joint distributions of multiple hook lengths and of frequencies and part sizes in partitions, extending prior work in this area. These results are discovered by investigating truncations of the…
Semigroup theory is a branch of abstract algebra, and it provides mathematical tools for the theory of computation. Finite semigroups can describe state transition systems and thus they model physically realizable computers. Engineering…
Given a positive integer k, we investigate the class of numerical semigroups verifying the property that every two subsequent non gaps, smaller than the conductor, are spaced by at least k. These semigroups will be called k-sparse and…
Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length $2$,…
The partition problem is a well-known basic NP-complete problem. We mainly consider the optimization version of it in this paper. The problem has been investigated from various perspectives for a long time and can be solved efficiently in…
Set partitions are arrangements of distinct objects into groups. The problem of listing all set partitions arises in a variety of settings, in particular in combinatorial optimization tasks. After a brief review, we give practical…
$\newcommand{\Arr}{\mathcal{A}} \newcommand{\numS}{k} \newcommand{\ArrX}[1]{\Arr(#1)} \newcommand{\eps}{\varepsilon} \newcommand{\opt}{\mathsf{o}}$ For point sets $P_1, \ldots, P_\numS$, a set of lines $L$ is halving if any face of the…
The aim of this paper is twofold. First, we study the number of partitions of a positive integer $m$ into at most $n$ parts in a given set $A$. We prove that such a number is bounded by the $n$-th Fibonacci number $F(n)$ for any $m$ and…