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We study the asymptotic distribution of the two following combinatorialparameters: the number of arc crossings in the linear representation, ${\mathrmcr^{(\ell)}$, and the number of chord crossings in the circular representation,${\mathrm cr^{(c)}$, of a random set partition. We prove that, for $k\leqn/(2\,\log n)$ (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, asymptoticallyGaussian as $n$ tends to infinity. We give exact and asymptotic formulas forthe variance of the distribution of the parameter ${\mathrm cr^{(\ell)}$ fromwhich we deduce that the distribution of ${\mathrm cr^{(\ell)}$ and ${\mathrmcr^{(c)}$ taken over all partitions of $[n]$ is concentrated around its mean.The proof of these results relies on a standard analysis of generatingfunctions associated with the parameter ${\mathrm cr^{(\ell)}$ obtained inearlier work of Stanton, Zeng and the author. We also determine the maximumvalues of the parameters ${\mathrm cr^{(\ell)}$ and ${\mathrm cr^{(c)}$.

On the limiting distribution of some numbers of crossings in set partitions