# A binary hook-length formula?

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.

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.