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 $p\ge 5$ is a prime number and $m=p_1^{{\alpha }_1}{p}_2^{{\alpha }_2}\cdots p_r^{{\alpha }_r},$ where ${\alpha }_i\ge 0$ and $p_i\ge 5$ 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}.

First draft of the paper. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2406.06224, arXiv:2406.07905