arXiv subjects

Maciej Zakarczemny

Publications and source records attributed to Maciej Zakarczemny.

8 recordsLinked to original sources

arXiv2026

The asymptotic behavior of the rectangle partition function $p(m,n)$

Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=π\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy--Ramanujan formula for integer partitions.

math.CO↗

arXiv2026

A note on the partition function of a rectangle

We investigate the asymptotic behavior of the rectangle partition functions $p(n,n)$ and $p(3,n)$. The function $p(n,n)$ counts partitions of the square $n\times n$ into rectangular blocks with integer sides, while $p(3,n)$ counts such partitions of the rectangle $3\times n$. Two rectangle partitions are identified when they contain the same multiset of rectangle types, and a block $a\times b$ is identified with a block $b\times a$. Our main results are $$ p(n,n) = \exp\left( \left(\tfracπ{\sqrt3}+o(1)\right)n\sqrt{\log n} \right) $$ and $$ p(3,n) = \exp\left( π\sqrt{\tfrac{11n}{3}}+O(\log n) \right). $$ We also present simpler upper and lower bounds for $p(n,n)$ and an independent upper bound for $p(3,n)$.

math.CO↗

arXiv2026

On the Diophantine Equation Involving Elementary Symmetric Polynomials and the Decomposition of Unity

We consider the equality of the values of the $n$th and $k$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. For $k < n$, we prove that this equation always has a solution, but only finitely many solutions. Furthermore, we consider the equality of the values of the $n$th and $(n-2)$th elementary symmetric polynomials of $n$ not necessarily distinct positive integers. In particular, we show that the number of solutions of this equation tends to infinity if $n$ tends to infinity.

math.NT↗

arXiv2025

Rectangle partitions generalizing integer partitions

In this paper, we introduce a natural geometric extension of the partition function. More precisely, we investigate the problem of counting partitions of a rectangle into rectangular blocks with integer sides. Here, two partitions of a rectangle are indistinguishable if they consist of the same multiset of blocks, their geometric arrangement does not matter.

math.CO↗

arXiv2024

Equal Sum and Product Problem III

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is twofold: first we present an asymptotic formula for $\sum\limits_{2\le n\le x}N(n)$, then we verify that the counting function $N(n)$ takes very large value compared to its average value.

math.NT↗

arXiv2022

On Frobenius problem with restrictions on common divisors of coefficients

Let $m,s,t$ are positive integers with $t\leq s-2$ and $a_1,a_2,\ldots,a_s$ are positive integers such that $(a_1,a_2,\ldots,a_{s-1})=1$. In the paper we prove that every sufficiently large positive integer can be written in the form $a_1μ_1+a_2μ_2+\ldots+a_sμ_m$, where positive integers $μ_1,μ_2,\ldots,μ_s$ have no common divisor being $m$-th power of a positive integer greater than $1$ but each $t$ of the values of $μ_1,μ_2,\ldots,μ_n$ have a common divisor being $m$-th power of a positive integer greater than $1$. Moreover, we show that every sufficiently large positive integer can be written as a sum of positive integers $μ_1,μ_2,\ldots,μ_n$ with no common divisor being $m$-th power of a positive integer greater than $1$ but each $s-1$ of the values of $μ_1,μ_2,\ldots,μ_s$ have a common divisor being $m$-th power of a positive integer greater than $1$.

math.NT↗

arXiv2019

On the zero-sum constant, the Davenport constant and their analogues

Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. For a positive integer $m$ (the case $m = 1$, is the classical one) let ${\mathsf E}_m(G)$ (or $η_m(G)$, respectively) be the least positive integer $t$ such that every sequence of length $t$ in $G$ contains $m$ disjoint zero-sum sequences, each of length $|G|$ (or of length $\le exp(G)$ respectively). In this paper, we prove that if $G$ is an~Abelian group, then ${\mathsf E}_m(G)=D(G)-1+m|G|$, which generalizes Gao's relation. We investigate also the non-Abelian case. Moreover, we examine the asymptotic behavior of the sequences $({\mathsf E}_m(G))_{m\ge 1}$ and $(η_m(G))_{m\ge 1}.$ We prove a~generalization of Kemnitz's conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the and we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.

math.AC↗

arXiv2019

Note on the Davenport's constant for finite abelian groups with rank three

Let $G$ be a finite abelian group and $D(G)$ denote the Davenport constant of $G$. We derive new upper bound for the Davenport constant for all groups of rank three. Our main result is that: $$D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})\le (n_1-1)+(n_2-1)+(n_3-1)+1+ (a_3-3)(n_1-1),$$ where $1<n_1|n_2|n_3\in\mathbb{N}$ and $a_3\le 20369$ is a constant. Therefore $D(C_{n_1}\oplus C_{n_2}\oplus C_{n_3})$ grows linearly with the variables $n_1,n_2,n_3.$ The new result is the given upper bound for $a_3$. Finally, we give an application of the Davenport constant to smooth numbers.

math.GR↗