An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a positive integer $s>1$.

Let $\mathcal{O}_{n,s}=\{\lambda\vdash n: \lambda_i\in\{1,3,5,\dots,2s-1\}\}$. Parts are odd integers.

Let $\mathcal{A}_{n,s}=\{\lambda\vdash n: a_1+2a_2+2a_3+\cdots+2a_s=n\,\,\, \text{and}\,\,\, a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

I would like ask:

>**QUESTION.** Is there a combinatorial or bijective proof of the equinumerosity $\#\mathcal{O}_{n,s}=\#\mathcal{A}_{n,s}$?