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

Let Bₔ, ₕ (n) denote the number of (u, v) -regular bipartitions of n. In this article, we prove that B₏, ₌ (n) is always almost divisible by p, where p 5 is a prime number and m=p₁^₁ p₂^₂ pᵣ^ᵣ, where ᵢ 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.