An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ 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.

Also, let $\mathcal{A}_{n,s}=\{a_1+2a_2+2a_3+\cdots+2a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

I would like ask:

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

Postscript. Fix $\ell>1$. Define $\mathcal{L}_{n,s}=\{\lambda\vdash n: \lambda_i\equiv 1\mod \ell\}$ and $\widetilde{\mathcal{L}}_{n,s}=\{a_1+\ell a_2+\ell a_3+\cdots+\ell a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

Then, Per Alexandersson's solution below appears to prove that $\#\mathcal{L}_{n,s}=\#\widetilde{\mathcal{L}}_{n,s}$.

An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ 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.

Also, let $\mathcal{A}_{n,s}=\{a_1+2a_2+2a_3+\cdots+2a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

I would like ask:

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

Postscript. Fix $\ell>1$. Define $\mathcal{L}_{n,s}=\{\lambda\vdash n: \lambda_i\equiv 1\mod \ell, \,\,i=1,\dots,s\}$ and $\widetilde{\mathcal{L}}_{n,s}=\{a_1+\ell a_2+\ell a_3+\cdots+\ell a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

Then, Per Alexandersson's solution below appears to prove that $\#\mathcal{L}_{n,s}=\#\widetilde{\mathcal{L}}_{n,s}$.

An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ 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.

Also, let $$\mathcal{A}_{n,s}=\{a_1+2a_2+2a_3+\cdots+2a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}\subset\mathbb{Z}^s_{\geq0}.$$

I would like ask:

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

An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ 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.

Also, let $\mathcal{A}_{n,s}=\{a_1+2a_2+2a_3+\cdots+2a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

I would like ask:

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

Postscript. Fix $\ell>1$. Define $\mathcal{L}_{n,s}=\{\lambda\vdash n: \lambda_i\equiv 1\mod \ell\}$ and $\widetilde{\mathcal{L}}_{n,s}=\{a_1+\ell a_2+\ell a_3+\cdots+\ell a_s=n: a_1\geq a_2\geq a_3\geq\cdots\geq a_s\geq0\}$.

Then, Per Alexandersson's solution below appears to prove that $\#\mathcal{L}_{n,s}=\#\widetilde{\mathcal{L}}_{n,s}$.