CONGRUENCES FOR BIPARTITIONS WITH ODD DESIGNATED SUMMANDS

Andrews, Lewis and Lovejoy investigated a new class of partitions with designated summands by taking ordinary partitions and tagging exactly one of each part size. Let $B_{2}(n)$ count the number of bipartitions of $n$ with designated summands in which all parts are odd. In this work, we establish many infinite families of congruences modulo powers of 2 and 3 for $B_{2}(n)$. For example, for each $n\geq 0$ and $\alpha \geq 0,$

\begin{equation*}

B_2\lb(48\cdot 5^{2\alpha+2}n+a_1\cdot 5^{2\alpha+1}\rb) \equiv 0 \pmod{9},

\end{equation*}

where $a_1 \in \{88, 136, 184, 232\}.$