Preparation: We consider the tensor-product vector space $(\mathbb{C}^2)^{\otimes n}$, where $\mathbb{C}^2$ is spanned by the normalized orthogonal basis $e_0$ and $e_1$. The tensor-product vector space $(\mathbb{C}^2)^{\otimes n}$ can be decomposed as Schur-duality; \begin{align*} (\mathbb{C}^2)^{\otimes n}=\bigoplus_{j=0 \hbox{ or } 1/2 }^{n/2} {\cal U}_{j} \otimes {\cal V}_j, \end{align*} where ${\cal U}_{j}$ expresses the spin $j$ representation of SU(2), and ${\cal V}_{j}$ expresses the irreducible representation of $n$-th permutation group with respect to the order of tensor product. That is, ${\cal V}_j$ is $2j+1$-dimensional space. We denote the projection to ${\cal U}_{j} \otimes {\cal V}_j$ by $P_{n,j}$. We choose the normalized vector $\Xi_{n,m}:= {n \choose m}^{-1/2} (e_1^{\otimes m} \otimes e_0^{\otimes n-m}+ PT)$, where $PT$ expresses permuted terms of $e_1^{\otimes m}\otimes e_0^{\otimes n-m}$. Then, we define the normalized vector \begin{align*} \Xi_{n,m|k,\ell}:= e_1^{\otimes \ell} \otimes e_0^{\otimes (k-\ell)}\otimes \Xi_{n-k,m-\ell} \in (\mathbb{C}^2)^{\otimes n}. \end{align*} We can show that $P_{n,j} \Xi_{n,m|k,\ell}= 0$ if and only if $\frac{n}{2}-j > \min(\ell,k-\ell,m,n-m) $.
Question: I would like to calculate the combinatorial quantity ${\langle\Xi_{n,m|k,\ell}, P_{n,j} \Xi_{n,m|k,\ell}\rangle}$ when $\frac{n}{2}-j \le \min(\ell,k-\ell,m,n-m) $.
When $j=n/2$, we find that $\langle\Xi_{n,m|k,\ell}, P_{n,j} \Xi_{n,m|k,\ell}\rangle= { \frac{m (m-1)\cdots (m-\ell+1) \cdot (n-m) (n-m-1)\cdots (n-m-(k-\ell)+1) }{ n (n-1)\cdots (n-k+1)} }$. Other cases are very difficult for me.