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 $\#\mathcal{O}\_{n,s}=\#\mathcal{A}\_{n,s}$?
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> 


**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 $\#\mathcal{L}\_{n,s}=\#\widetilde{\mathcal{L}}\_{n,s}$.



Seeking a bijective proof enumerating two partition sets: Part I
Take a partition (Young diagram) in $O\_{n,s}$.
Number of elements in first column is $a\_1$.
Removing the first column now gives a partition with only even parts.
Divide all parts by $2$, and take conjugate. This gives $a\_2 \geq a\_3 \geq \dotsc \geq a\_s$, and you now have an element in the second set.


*It is a nice problem, I will most likely steal it!*


