New Congruences of Partitions With Odd Parts Distinct

Abstract

Let $\mathrm{pod}(n)$ denote the number of partitions of $n$ with odd parts distinct, and ${{r}_{k}}(n)$ be the number of representations of $n$ as sum of $k$ squares. We find the following two arithmetic relations: for any integer $n\ge 0$, \[\mathrm{pod}(3n+2)\equiv 2{{(-1)}^{n+1}}{{r}_{5}}(8n+5) \pmod{9}, \] and \[\mathrm{pod}(5n+2)\equiv 2{{(-1)}^{n}}{{r}_{3}}(8n+3) \pmod{5}.\] From which we deduce many interesting congruences including the following two infinite families of Ramanujan-type congruences: for $a \in \{11, 19\}$ and any integers $\alpha \ge 1$ and $n \ge 0$, we have \[\mathrm{pod}\Big({{5}^{2\alpha +2}}n+\frac{a \cdot {{5}^{2\alpha +1}}+1}{8}\Big)\equiv 0 \pmod{5}.\]