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:
$$E\_{\lambda\nu}(N)= \frac{1}{(m-n)!}{m-n \choose m-n-\ell(\nu)+\ell(\lambda)}[N]\_{\nu/\lambda}.$$



Sum involving determinants of binomial coefficients, indexed by partitions
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..


