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

Abstract

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 $k\le n/(2\log n)$ (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 ${\mathrm cr^{(c)}$.

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• Average values of some Z-parameters in a random set partition