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


Also, let $\mathcal{A}\_{n,s}=\{a\_1+2a\_2+2a\_3+\dotsb+2a\_s=n: a\_1\geq a\_2\geq a\_3\geq\dotsb\geq a\_s\geq0\}\subset\mathbb{Z}^s\_{\geq0}$. In my earlier MO post [Seeking a bijective proof enumerating two partition sets: Part I](https://mathoverflow.net/questions/426984/seeking-a-bijective-proof-enumerating-two-partition-sets-part-i), I proposed a question on
$\#\mathcal{O}\_{n,s}=\#\mathcal{A}\_{n,s}$ which was subsequently [answered](https://mathoverflow.net/a/426990) by Per Alexandersson.


Let's add one more set of partitions
$$\mathcal{B}\_{n,s}=\{a\_1+a\_2+\cdots+a\_s=n: a\_2\geq 2a\_1, 2a\_3\geq 3a\_2, 3a\_4\geq 4a\_3,\dotsc,(s-1)a\_s\geq sa\_{s-1}\}.$$
I would like ask:



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



Seeking a bijective proof enumerating two partition sets: Part II
Just to mark this question as answered, let me convert my comments into an answer.


The set of partitions $\mathcal{B}\_{n,s}$ you define are called the "lecture hall partitions" and they were introduced by Mireille Bousquet-Mélou and Kimmo Eriksson in [1]. See also the survey on the mathematics of lecture hall partitions by Carla Savage in [2]. The fact that there are the same number of lecture hall partitions of $n$ into at most $s$ parts as partitions of $n$ into odd parts less than $2s$ was the main result of that first paper of Bousquet-Mélou and Eriksson, and they gave a couple of different proofs: one based on affine Coxeter groups, the other more direct and combinatorial. At the end of their paper they gave a recursive bijection between $\mathcal{B}\_{n,s}$ and $\mathcal{O}\_{n,s}$ based on their combinatorial argument. A simple bijection was later obtained by Niklas Eriksen; see the FPSAC 2002 conference article [3].


[1] *Bousquet-Mélou, Mireille; Eriksson, Kimmo*, [**Lecture hall partitions**](http://dx.doi.org/10.1023/A:1009771306380), Ramanujan J. 1, No. 1, 101-111 (1997). [ZBL0909.05008](https://zbmath.org/?q=an:0909.05008).


[2] *Savage, Carla D.*, [**The mathematics of lecture hall partitions**](http://dx.doi.org/10.1016/j.jcta.2016.06.006), J. Comb. Theory, Ser. A 144, 443-475 (2016). [ZBL1343.05032](https://zbmath.org/?q=an:1343.05032).


[3] *Eirksen, Niklas*, **[A simple bijection between lecture hall partitions and partitions into odd integers](http://igm.univ-mlv.fr/~fpsac/FPSAC02/ARTICLES/Eriksen2.pdf)**, [FPSAC 2002](https://igm.univ-mlv.fr/~fpsac/FPSAC02/articles.html).


