On $(4, 5)$-regular bipartitions with odd parts distinct

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In his work, K. Alladi considered the partition function $pod(n)$, the number of partitions of an integer $n$ with odd parts distinct (the even parts are unrestricted). He obtained a series expansion for the product generating function of these partitions. Later Hirschhorn and Sellers obtained some internal congruences involving the infinite families and Ramanujan's type congruences for $pod(n)$. Let $B_{4, 5}(n)$ denote the number of $(4, 5)$-regular bipartitions of a positive integer $n$ with odd parts distinct. In this paper, we establish many infinite families of congruences modulo powers of $2$ for $B_{4, 5}(n)$.