The Kostka number $K_{\lambda \mu}$ gives the number of SSYT of shape $\lambda$ with content $\mu$. Are there any results regarding Kostka numbers for "partial contents" (I do not know if there is a term for this already used in the literature)? By this, I mean something like "How many SSYT exist of shape $\lambda$ with exactly $a_1$ 1 entries, $a_2$ 2 entries, and any number of entries from $\{3, \ldots, n\}$?" This would be a sum of Kostka numbers, something like $$\tilde{K}_{\lambda (a_1, a_2)} = \sum_{\substack{x_3, \ldots, x_n \\ \mu = (a_1, a_2, x_3, \ldots, x_n) \\ a_1 + a_2 + x_3 + \dots + x_n = |\lambda|}} K_{\lambda \mu}\,.$$
Any results in this direction would be very helpful for computing certain coefficients that appear in the anomaly cancellation equations for 6D supergravity theories.
Edit: As mentioned in the comment by Jair Taylor, these could also be expressed in terms of skew Kostka numbers, something like $$\tilde{K}_{\lambda (a_1, a_2)} = \sum_{\substack{\mu \\ |\mu| = a_1 + a_2}} K_{\mu (a_1, a_2)} \sum_{\substack{\gamma \\ |\gamma| = |\lambda| - a_1 - a_2}} K_{\nu / \mu, \gamma}\,.$$ To me, this seems somewhat more involved than the original sum, but it could be simplified if there is some analogue of the hook-content formula for skew tableaux.
Let $\mu'$ be the partition $1^{n_1}2^{n_2}\cdots l^{n_l}$ with $|\mu'| = k$. Any SSYT of shape $\lambda$ with $n_1$ $1$'s, $n_2$ $2$'s, etc. and any number of $l+1$'s, $l+2$'s, ... is given by
1) an SSYT of some shape $\lambda'$ and content $\mu'$, and
2) an skew SSYT of shape $\lambda / \lambda'$.
Hence the number of $SSYT$'s satisfying these conditions is
$$\sum_{\lambda' \vdash k} K_{\lambda', \mu'} \sum_{\nu} K_{\lambda / \lambda', \nu}$$
The sum $$\sum_{\nu} K_{\lambda / \lambda', \nu}$$ is the skew Schur function $s_{\lambda/\lambda'}$ evaluated at $x_1 = 0, \ldots, x_l = 0, x_{l+1} = 1, x_{l+2} =1, \ldots, x_{n} = 1$; or equivalently, $x_1 = x_2 = \ldots = x_{n-l} = 1$, $x_{n-l+1} = \cdots = 0$. Hence your the your count is
$$\sum_{\lambda' \vdash k} K_{\lambda', \mu'} s_{\lambda/\lambda'}(1, 1, \ldots, 1).$$
This should be somewhat faster to compute than your count, since there are various ways of evaluating Schur function such as the Jacobi-Trudi identity. It may be possible to simplify this more - I am not sure.