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

#### Print ISSN: 0972-7752 | Online ISSN: 2582-0850

**Abstract**

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*}

**Keywords and Phrases**

Partition identities, Theta--functions, Partition congruences, Regular partition.

**A.M.S. subject classiﬁcation**

11P83, 05A17.

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