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.
Best Answer
Let $k(\lambda)=\sum_\mu K_{\lambda\mu}$. Then we have the generating function $\sum_\lambda k(\lambda)s_\lambda = \prod_{n\geq 1} (1-h_n)^{-1}$.