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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 $\lambda$, let $Y_{\lambda}$ be the corresponding Young diagram. If $\square$ a cell in $Y_{\lambda}$, construct its hook-length $h_{\square}$ 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.

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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.

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