Distribution and congruences of ℓ-regular bipartitions

Let B-& ell;(n) denote the number of & ell;-regular bipartitions of n. In 2013, Lin [6] proved a density result for B-4(n). He showed that for any positive integer k, B-4(n) is almost always divisible by 2(k). In this article, we significantly extend his result. We prove that B-2 alpha(m)(n) and B-3 alpha(m)(n) are almost always divisible by arbitrary power of 2 and 3 respectively. Further, we obtain an infinite family of congruences and internal congruences for B-2(n) and B-4(n) by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi [14] on nilpotency of Hecke operator, we prove that there exists an infinite family of congruences modulo arbitrary power of 2 satisfied by B-2 alpha(n).