Ttdfjt

This is purely exploratory and inspired by curiosity.

Setup: For an integer $k>0$, let $k=sum_jgeq0k_j2^j$ be its binary expansion and denote the sum of its digits by $eta(k):=sum_jk_j$. Further, introduce a binary factorial
$$[n]!_b:=eta(1)eta(2)cdotseta(n).$$
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 $h_square:squarein Y_lambda=5,3,1,3,1,1$ which shall be replaced by $eta(h_square):squarein Y_lambda=2,2,1,2,1,1$.

Naturally, we define the (new) product of hook-lengths and denote (with an abuse of notation)
$$H_lambda=prod_squareinlambdaeta(h_square).$$
If $lambdavdash n$, it is easy to verify that $frac[n]!_bH_lambda$ is an integer.

QUESTION. What do these integer count?
$$sum_lambdavdash nfrac[n]!_bH_lambda.$$

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 $N=2pchoose p+1$. And $[2n+1]!_b$ by $p^K$ for $K=2p+1choose p+2p+1<2N$. Then for $2times N$ diagram we get $p$ in the denominator.