This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.


Suppose that $n=k+l+r$ where $k\geq l\geq r\geq 0$. Let $G$ be the symmetric group $S\_n$ and $H$ be its Young subgroup $S\_k \times S\_l \times S\_1^r$. What can be said about the induced representation $\mbox{Ind}\_H^G 1$? What are its components? Can we say something about their multiplicities?



Induced representation of a Young subgroup
The answer is a special case of Young's rule. In [my book](http://www.imsc.res.in/~amri/rtcv.html), I give a very simple method for the slightly easier case where $r=0$. In that case we have:
$$
\mathrm{Ind}\_{S\_k\times S\_l}^{S\_n} = \bigoplus\_{s=0}^l V\_{(n-s, s)},
$$
a multiplicity-free decomposition, where $V\_{(n-s, s)}$ is an irreducible representation of dimension $\binom ns - \binom n{s-1}$.


In your more general case, the irreducible constituents will all be Specht modules of the form $V\_{(s, t, u)}$, where $s\geq k$, $s+t\geq k+l$, and (of course) $s+t+u=k+l+r$. The multiplicities are given by [Kostka numbers](https://en.wikipedia.org/wiki/Kostka_number), which can be greater than one. For example, if $(k, l, r) = (3, 2, 1)$, then $V\_{(5, 1)}$ occurs with multiplicity $2$.


The multiplicity of $V\_{(s, t, u)}$ with $(s, t, u)=(5, 1, 0)$ can be calculated in [sage](http://www.sagemath.org) as, for example:



```
sage: sage.libs.symmetrica.all.kostka_number([5, 1], [3, 2, 1])
2

```

More generally, replace `[5, 1]` with `[s, t, u]`. 


