Let $s\_{\lambda}$ and $m\_{\lambda}$ be the Schur and monomial symmetric functions indexed by an integer partition $\lambda$ ($\ell(\lambda)$ is the number of parts of $\lambda$ and $m\_i(\lambda)$ is the multiplicity of part $i$). By the hook-content formula we have:
$$
s\_{\lambda}(1^n) = \prod\_{u\in \lambda} \frac{n+c(u)}{h(u)},
$$
where $c(u)$ and $h(u)$ are the content and hook length of the cell $u\in \lambda$.


Using $s\_{\lambda} = \sum\_{\mu} K\_{\lambda \mu} m\_{\mu}$ where $K\_{\lambda \mu}$ is the Kostka number, the number of semistandard Young tableaux of shape $\lambda$ and type $\mu$. Then we get $\sum\_{\mu} K\_{\lambda \mu} m\_{\mu}(1^n)=\prod\_{u\in \lambda} \frac{n+c(u)}{h(u)}$. This counts semistandard Young tableaux of shape $\lambda$ and any type.


Does the sum $\sum\_{\mu}K\_{\lambda,\mu}$ have a known formula for $\ell(\lambda)\geq 2$? This would be the number of semistandard Young tableaux of shape $\lambda$ with partition type.



Is there a known formula for the number of SSYT of given shape with partition type?
Let $k(\lambda)=\sum\_\mu K\_{\lambda\mu}$. Then we have the generating function
$\sum\_\lambda k(\lambda)s\_\lambda = \prod\_{n\geq 1} (1-h\_n)^{-1}$. 


