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Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
S. Sriram, A. David Christopher
https://arxiv.org/abs/2509.26559

Congruences Relating Regular Partition Functions, a Genearalised Tau Function and Partition Function Weighted Composition Sums
Let $n$ and $t$ be positive integers with $t\geq 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted $τ_k(n)$, is defined as follows: \[q\prod_{m=1}^{\infty}(1-q^m)^k=\sum_{n=1}^{\infty}τ_k(n)q^n, \] where $k$ is an integer. We express $τ_k(n)$ as a binomial coefficient weighted partition sum. Consequently, we obtain congruence identities that relate $τ_k(n)$, $R_t(n)$ and partition function weighted composition sums.