This is answered in Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley.

You can give a formula for these numbers in terms of plethysm and internal product and inner product.

The starting point is Exercise 7.74 of Enumerative Combinatorics II by Richard Stanley which is a formula for the Schur functors of the defining representation of the symmetric group in terms of plethysm and inner product.