The answer is a special case of Young's rule. In my book, 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, 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 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]`

.