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

We study the asymptotic distribution of the two following combinatorial parameters: the number of arc crossings in the linear representation, ${\mathrm cr^{(\ell)}$,andthenumberofchordcrossingsinthecircularrepresentation,${\mathrm cr^{©}$,ofarandomsetpartition\mathrm{.}Weprovethat,for$k\leq n/(2,\log n)$(resp\mathrm{.},${k=o(\sqrt{n})}$),thedistributionoftheparameter${\mathrm cr^{(\ell)}$(resp\mathrm{.},${\mathrm cr^{©}$)takenoverpartitionsof$[n]:={1,2,…,n}$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 ${\mathrm cr^{©}$takenoverallpartitionsof$[n]$ 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)}.