Distribution and congruences of $(u,v)$-regular bipartitions

Abstract

Let $B_{u,v}(n)$ denote the number of $(u,v)$-regular bipartitions of $n$. In this article, we prove that $B_{p,m}(n)$ is always almost divisible by $p,$ where $p\geq 5$ is a prime number and $m=p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_r^{\alpha_r}, $ where $\alpha_i \geq 0$ and $p_i \geq 5$ be distinct primes with $\gcd(p,m)=1$ . Further, we obtain an infinities families of congruences modulo $3$ for $B_{3,7}(n),$ $B_{3,5}(n)$ and $B_{3,2}(n)$ by using Hecke eigenform theory and a result of Newman \cite{Newmann1959}. Furthermore, we get many infinite families of congruences modulo $7$, $11$ and $13$ respectively for $B_{2,7}(n)$, $B_{2,11}(n)$ and $B_{2,13}(n),$ by employing an identity of Newman \cite{Newmann1959}. In addition, we prove infinite families of congruences modulo $2$ for $B_{4,3}(n)$, $B_{8,3}(n)$ and $B_{4,5}(n)$ by applying another result of Newman \cite{Newmann1962}.