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

Let Bu,v(n) denote the number of $(u,v)$-regular bipartitions of n. In this article, we prove that Bp,m(n) is always almost divisible by $p,$ where p≥ 5 is a prime number and m=p₁α₁ p₂α₂⋯ pᵣαᵣ, where αᵢ ≥ 0 and pᵢ ≥ 5 be distinct primes with (p,m)=1 . Further, we obtain an infinities families of congruences modulo $3$ for B3,7(n), B3,5(n) and B3,2(n) by using Hecke eigenform theory and a result of Newman {Newmann1959}. Furthermore, we get many infinite families of congruences modulo $7$, $11$ and $13$ respectively for B2,7(n), B2,11(n) and B2,13(n), by employing an identity of Newman {Newmann1959}. In addition, we prove infinite families of congruences modulo $2$ for B4,3(n), B8,3(n) and B4,5(n) by applying another result of Newman {Newmann1962}.