Probability that two random permutations of an $n$-set commute? From this MathOverflow question:

It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin Steinberg

Unfortunately, it's not well known to me.  Can I get a reference or link to this result?  Or a proof, if it's simple enough.
(Google doesn't work where I am; I tried Binging two random permutations commute but it only gives the MathOverflow link.)
I need this result for a secret sharing scheme I'm currently analyzing.

Just to add a bit more detail to blue's answer, for a finite group $G$, the probability that two randomly chosen elements commute is 
$$\frac{1}{|G|^2}\sum_{g \in G} |C_G(g)| = \frac{1}{|G|}\sum_{g \in G} \frac{1}{|{\rm Cl}(g)|} = \frac{k}{|G|}$$
where $k$ is the number of conjugacy classes of $G$.
