Equal values of certain partition functions via Diophantine equations

TL;DR

This paper investigates the existence of integer solutions to Diophantine equations involving partition functions, specifically when two different sets produce equal partition counts for some integers.

Contribution

It establishes new results on the conditions under which equations of the form P_A(x)=P_B(y) have solutions, linking partition functions and Diophantine equations.

Findings

Identifies conditions for solutions to P_A(x)=P_B(y)

Provides examples of sets A, B with equal partition values

Advances understanding of partition function equalities

Abstract

Let $A\subset \N_{+}$ and by $P_{A}(n)$ denotes the number of partitions of an integer $n$ into parts from the set $A$. The aim of this paper is to prove several result concerning the existence of integer solutions of Diophantine equations of the form $P_{A}(x)=P_{B}(y)$, where $A, B$ are certain finite sets.