Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums

Let $n$ and $t$ be positive integers with $t\ge 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted ${\tau }_k(n)$, is defined as follows: $$q\prod _{m=1}^{\mathrm{\infty }}(1-q^m{)}^k=\sum _{n=1}^{\mathrm{\infty }}{\tau }_k(n)q^n,$$ where $k$ is an integer. We express ${\tau }_k(n)$ as a binomial coefficient weighted partition sum. Consequently, we obtain congruence identities that relate ${\tau }_k(n)$, $R_t(n)$ and partition function weighted composition sums.