  We study the asymptotic distribution of the two following combinatorial
parameters: the number of arc crossings in the linear representation, ${\mathrm
cr^{(\ell)}$, and the number of chord crossings in the circular representation,
${\mathrm cr^{(c)}$, of a random set partition. We prove that, for $k\leq
n/(2\,\log n)$ (resp., ${k=o(\sqrt{n})}$), the distribution of the parameter
${\mathrm cr^{(\ell)}$ (resp., ${\mathrm cr^{(c)}$) taken over partitions of
$[n]:=\{1,2,...,n\}$ into $k$ blocks is, after standardization, asymptotically
Gaussian as $n$ tends to infinity. We give exact and asymptotic formulas for
the variance of the distribution of the parameter ${\mathrm cr^{(\ell)}$ from
which we deduce that the distribution of ${\mathrm cr^{(\ell)}$ and ${\mathrm
cr^{(c)}$ taken over all partitions of $[n]$ is concentrated around its mean.
The proof of these results relies on a standard analysis of generating
functions associated with the parameter ${\mathrm cr^{(\ell)}$ obtained in
earlier work of Stanton, Zeng and the author. We also determine the maximum
values of the parameters ${\mathrm cr^{(\ell)}$ and ${\mathrm cr^{(c)}$.
