Related papers: A combinatorial problem and numerical semigroups

Can a list of binary strings be ordered so that consecutive strings differ in a single bit? Can a list of permutations be ordered so that consecutive permutations differ by a swap? Can a list of non-crossing set partitions be ordered so…

We investigate compositions of a positive integer with a fixed number of parts, when there are several types of each natural number. These compositions produce new relationships among binomial coefficients, Catalan numbers, and numbers of…

A positive integer $n$ is said to be a Zumkeller number or an integer-perfect number if the set of its positive divisors can be partitioned into two subsets of equal sums. In this paper, we prove several results regarding Zumkeller numbers.…

This paper investigates the exponential Diophantine equation of the form $a^x+b=c^y$, where $a, b, c$ are given positive integers with $a,c \ge 2$, and $x,y$ are positive integer unknowns. We define this form as a "Type-I transcendental…

Let $A,B,C,D$ be rational numbers such that $ABC \neq 0$, and let $n_1>n_2>n_3>0$ be positive integers. We solve the equation $$ Ax^{n_1}+Bx^{n_2}+Cx^{n_3}+D = f(g(x)),$$ in $f,g \in \mathbb{Q}[x]$. In sequel we use Bilu-Tichy method to…

Given two sets of positive integers $A$ and $B$, let $AB := \{ab : a \in A,\, b \in B\}$ be their product set and put $A^k := A \cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every…

Given a constraint satisfaction problem (CSP) on $n$ variables, $x_1, x_2, \dots, x_n \in \{\pm 1\}$, and $m$ constraints, a global cardinality constraint has the form of $\sum_{i = 1}^{n} x_i = (1-2p)n$, where $p \in (\Omega(1), 1 -…

We study pairs and m--tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of…

Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The…

We say the sets of nonnegative integers A and B are additive complements if their sum contains all sufficiently large integers. In this paper we prove a conjecture of Chen and Fang about additive complement of a finite set.

Given a sequence of $N$ positive real numbers $\{a_1,a_2,..., a_N \}$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is…

Let $C\subset\mathbb{N}^p$ be an integer polyhedral cone. An affine semigroup $S\subset C$ is a $ C$-semigroup if $| C\setminus S|<+\infty$. This structure has always been studied using a monomial order. The main issue is that the choice of…

This paper deals with the following question: Suppose that there exist an integer or a non-negative integer solution $x$ to a system $Ax = b$, where the number of non-zero components of $x$ is $n$. The target is, for a given natural number…

To factor an integer N, given that it is equal to the product of two primes, it suffices to find an integer d satisfying a certain simple numerical test. In this approach, the factorization problem equates to the problem of designing an…

Let $C$ be the middle-third Cantor set. We show that \[\left\{\frac{1}{n!}: n\in\mathbb{N}\right\}\cap C=\left\{1, \frac{1}{5!}\right\}.\] This answers a question recently posed by Jiang [J. Lond. Math. Soc., 2026, published online]. Our…

In the Categorical Clustering problem, we are given a set of vectors (matrix) A={a_1,\ldots,a_n} over \Sigma^m, where \Sigma is a finite alphabet, and integers k and B. The task is to partition A into k clusters such that the median…

A combinatorial structure, $\mathcal{F}$, with counting sequence $\{a_n\}_{n\ge 0}$ and ordinary generating function $G_\mathcal{F}=\sum_{n\ge0} a_n x^n$, is positive algebraic if $G_\mathcal{F}$ satisfies a polynomial equation…

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is…

In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed…

Let $n$ be a positive integer. A collection $\cal S$ of subsets of $[n]=\{1,\ldots,n\}$ is called {\it symmetric} if $X\in {\cal S}$ implies $X^\ast\in {\cal S}$, where $X^\ast:=\{i\in [n]\colon n-i+1\notin X\}$. We show that in each of the…