co.combinatorics - Is there a known formula for the number of SSYT of given shape with partition type?

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:
$$
s_{lambda}(1^n) = prod_{uin lambda} frac{n+c(u)}{h(u)},
$$
where $c(u)$ and $h(u)$ are the content and hook length of the cell $uin lambda$.

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 $sum_{mu} K_{lambda mu} m_{mu}(1^n)=prod_{uin lambda} frac{n+c(u)}{h(u)}$. 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.