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 ${ \mathrm cr^{ (c)}$.
Categories: math.CO