TITLE: Sum involving determinants of binomial coefficients, indexed by partitions
QUESTION [5 upvotes]: I would appreciate some help proving a conjecture related to combinatorics and representation theory.
Given an integer partition $\lambda\vdash n$, define a polynomial in $N$ whose roots are the negatives of the contents of the partition,
$$ [N]_\lambda=\prod_{\square \in \lambda}(N+c(\square)).$$ This polynomial is closely related to the value of a Schur function evaluated at the $N\times N$ identity matrix. On the other hand, given $\nu\vdash m$ and $\rho\vdash k$ contained in $\nu$, Jacobi-Trudi applied to a skew-Schur function leads to a determinant of binomial coefficients
$$s_{\nu/\rho}(1_N)=\det_{1\le,i,j\le m}\left({N+\nu_i-i-\rho_j+j-1 \choose \nu_i-i-\rho_j+j}\right).$$ The final ingredient I need for my question is another determinant of binomials,
$$A_{\lambda\rho}=\det_{1\le,i,j\le k}\left({\rho_i-i \choose \lambda_j-j}\right).$$
Now, in the course of some physics calculation, I arrived at the quantity
$$ E_{\lambda\nu}(N)=\sum_{\lambda\subset\rho\subset\nu} A_{\lambda\rho}s_{\nu/\rho}(1_N).$$
I thought this was as far as I could push it, but experimentation convinced me that, as a function of $N$, this guy satisfies
$$ E_{\lambda\nu}(N)\propto [N]_{\nu/\lambda}.$$
It is very surprising to me that this sum should factor like this.
The question is how to prove the above conjecture.
For example, if $\nu=(2,2,1)$ and $\lambda=(1)$, the six terms in the sum are
$$\{\frac{1}{24}N(N^2-1)(5N-6),-\frac{1}{2}N^2(N-1) ,\frac{1}{3}N(N^2-1) ,\frac{1}{2}N(N-1),-N^2,N\}.$$ When all these are added, the result is proportional to $N(N-2)(N^2-1)=[N]_{(2,2,1)/(1)}$.
Actually, I think I know the proportionality constant when $\nu$ and $\lambda$ are both hooks:
$$E_{\lambda\nu}(N)= \frac{1}{(m-n)!}{m-n \choose m-n-\ell(\nu)+\ell(\lambda)}[N]_{\nu/\lambda}.$$

REPLY [2 votes]: Following the OP's comment that the factorization can be obtained using my comments above, I repost them here.
The quantity $E_{\lambda\nu}(N)$ can be computed using Cauchy-Binet. Specifically see Lemma 9.1 in Yeliussizov's (nice) paper: https://arxiv.org/pdf/1601.01581.pdf.
For the quantity $A_{\lambda\rho}$, check equation 53 in loc.cit..