What is the number of semi-standard tableau (weakly increasing on rows and strictly increasing on columns) for the partition $2n=n+n$ with entries $\{1,2, \cdots ,n\}$ such that each $i$ appears exactly twice? I guess it has something to do with two copies of a irreducible representation of the general linear group $GL\_n$ and I have tried computing this using Hook length formula but failed subsequently. And, is there a way to get all of them? 



Number of semi-standard tableau
You want the coefficient of $(x\_1x\_2\cdots x\_n)^2$ in the schur polynomial associated to the partition $(n,n)$. By Jacobi-Trudi this can be written in terms of the complete homogeneous symmetric functions as
$$s\_{(n,n)}=\begin{vmatrix} h\_n & h\_{n+1} \\ h\_{n-1} & h\_n\end{vmatrix}.$$
From here you can find that this quantity is the coefficient of $t^n$ in $(1-t)(1+t+t^2)^n$. These numbers are called [Riordan numbers](https://oeis.org/A005043). Their generating function is given by
$$\frac{2}{1+x+\sqrt{1-2x-3x^2}},$$
and they satisfy several recurrence relations that you can find in that OEIS link.


