CONGRUENCES FOR (4, 5)-REGULAR BIPARTITIONS INTO DISTINCT PARTS

Let $B_{4, 5}(n)$ denote the number of $(4, 5)$-regular bipartitions of a positive integer $n$ into distinct parts. In this paper, we establish many infinite families of congruences modulo powers of $2$ for $B_{4, 5}(n)$. For example,

\begin{align*}

\nonumber &\sum_{n=0}^{\infty}B_{4, 5}\lb(16\cdot 3^{2\alpha}\cdot 5^{2\beta}\cdot 7^{2\gamma} n+2\cdot 3^{2\alpha}\cdot 5^{2\beta}\cdot 7^{2\gamma}-1\rb)q^n \\& \equiv 2f_1^3 \pmod{4}, \, \text{for all}\,\, \alpha, \beta, \gamma \geq 0. \end{align*}