New Congruences of Partitions With Odd Parts Distinct

Let $\mathrm{p}\mathrm{o}\mathrm{d}(n)$ 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 $n\ge 0$, $$\mathrm{p}\mathrm{o}\mathrm{d}(3n+2)\equiv 2{(-1)}^{n+1}{r}_5(8n+5)(\mathrm{m}\mathrm{o}\mathrm{d}9),$$ and $$\mathrm{p}\mathrm{o}\mathrm{d}(5n+2)\equiv 2{(-1)}^n{r}_3(8n+3)(\mathrm{m}\mathrm{o}\mathrm{d}5)\mathrm{.}$$ 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 $$\mathrm{p}\mathrm{o}\mathrm{d}(5^{2\alpha +2}n+\frac{a\cdot 5^{2\alpha +1}+1}{8})\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}5)\mathrm{.}$$