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

TL;DR

This paper investigates the asymptotic distribution of arc and chord crossings in random set partitions, showing they are approximately Gaussian under certain conditions and providing formulas for their variance.

Contribution

It establishes the asymptotic Gaussian behavior of crossing parameters in set partitions and derives variance formulas, extending previous combinatorial analyses.

Findings

Crossing counts are asymptotically Gaussian for large n.

Variance formulas for crossing parameters are provided.

Distribution concentrates around the mean for large 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 ${\mathrm cr^{(c)}$.