Decomposition of Representations of $S_n$
Solution 1:
You appear to want the multiplicities of Specht modules in the permutation module. These are given by Young's rule, and appears in all standard sets of lecture notes and books about symmetric groups. The coefficients $m_{\lambda,\mu}$ in the decomposition are equal to the number of semistandard tableaux of shape $\lambda$ and type $\mu$, and are called the Kostka numbers.
Wikipedia has a write-up about it.