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A. Kasraoui

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

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