Arithmetic density and congruences of $\mathrm{\ell }$-regular bipartitions $II$

Let $B_{\mathrm{\ell }}(n)$ denote the number of $\mathrm{\ell }-$regular bipartitions of $n\mathrm{.}$ In 2013, Lin \cite{Lin2013} proved a density result for $B_4(n)\mathrm{.}$ He showed that for any positive integer $k,$ $B_4(n)$ is almost always divisible by $2^k\mathrm{.}$ In this article, we improved 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 infinities families of congruences and multiplicative formulae for $B_2(n)$ and $B_4(n)$ by using Hecke eigenform theory. Next, by using a result of Ono and Taguchi on nilpotency of Hecke operator, we also find an infinite families of congruences modulo arbitrary power of $2$ satisfied by $B_{2^{\alpha }}(n)\mathrm{.}$