# Sum involving determinants of binomial coefficients, indexed by partitions

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

• What is the actual question then in this case? It seems no "formal" question has been stated in the post.... – Suvrit 2 days ago
• An obvious but probably unhelpful suggestion is to try to put variables $x_i$ back into the equation. – Sam Hopkins 2 days ago
• Very vague comment. I wonder if your $E_{\lambda,\nu}$ comes from Cauchy-Binet applied to a minor of an appropriate product of matrices. Skew Schurs can be recognized as minors of a matrix (think Jacobi-Trudi). I think your $A_{\lambda\rho}$s can also be recognized as minors of an appropriate infinite matrix. Perhaps the product of matrices has nice structure. – user61318 3 hours ago
• If I didn't mess up, $A_{\lambda\rho}$ counts Molev's dual hook tableaux with shape $\lambda/\rho$ . Following Sam Hopkins' suggestion to put variables (two sets even) back might be good. – user61318 1 hour ago
• @user61318 I have never heard about Molev's dual hook tableaux. Where can I learn about them? – Marcel 20 mins ago