This is purely exploratory and inspired by curiosity.


**Setup:** For an integer $k>0$, let $k=\sum\_{j\geq0}k\_j2^j$ be its binary expansion and denote the sum of its digits by $\eta(k):=\sum\_jk\_j$. Further, introduce a *binary factorial*
$$[n]!\_b:=\eta(1)\eta(2)\cdots\eta(n).$$ 
Given an [integer partition](https://en.wikipedia.org/wiki/Partition_(number_theory)) $\lambda$, let $Y\_{\lambda}$ be the corresponding Young diagram. If $\square$ a cell in $Y\_{\lambda}$, construct its [hook-length](https://en.wikipedia.org/wiki/Hook_length_formula) $h\_{\square}$ in the usual manner but replace it by $\eta(h\_{\square})$. 


For example, take $\lambda=(3,2,1)$ then its multiset of hooks is $\{h\_{\square}:\square\in Y\_{\lambda}\}=\{5,3,1,3,1,1\}$ which shall be replaced by $\{\eta(h\_{\square}):\square\in Y\_{\lambda}\}=\{2,2,1,2,1,1\}$.


Naturally, we define the (new) product of hook-lengths and denote (with an abuse of notation)
$$H\_{\lambda}=\prod\_{\square\in\lambda}\eta(h\_{\square}).$$
If $\lambda\vdash n$, it is easy to verify that $\frac{[n]!\_b}{H\_{\lambda}}$ is an integer.



> 
> **QUESTION.** What do these integer count?
>  $$\sum\_{\lambda\vdash n}\frac{[n]!\_b}{H\_{\lambda}}.$$
> 
> 
> 


The first few values are: $1, 2, 3, 7, 10, 23, 52, 82, 117, 258, \dots$ but not listed on [OEIS](https://oeis.org/).



A binary hook-length formula?
I am afraid they are not always integers. Take large $p$ and $n=2^{2p}-1$. Then $[n]!\_b$ is divisible by $p^N$ for $N={2p\choose p}+1$. And $[2n+1]!\_b$ by $p^K$ for $K={2p+1\choose p}+2p+1<2N$. Then for $2\times N$ diagram we get $p$ in the denominator. 


