Attacks and alignments: rooks, set partitions, and permutations

TL;DR

This paper studies the asymptotic distribution of component sizes in random set partitions and permutations with a fixed number of blocks or cycles, revealing Poisson limits for size-3 components under a specific regime.

Contribution

It introduces a novel combinatorial approach using rook placements and the Chen--Stein method to analyze component size distributions in these combinatorial structures.

Findings

Number of size-3 components converges to Poisson distribution with mean 2/3 t^2 for set partitions.

Number of size-3 components converges to Poisson distribution with mean 4/3 t^2 for permutations.

All other components are predominantly of size one or two with high probability.

Abstract

We consider uniformly random set partitions of size $n$ with exactly $k$ blocks, and uniformly random permutations of size $n$ with exactly $k$ cycles, under the regime where $n-k \sim t\sqrt{n}$, $t>0$. In this regime, there is a simple approximation for the entire process of component counts; in particular, the number of components of size 3 converges in distribution to Poisson with mean $\frac{2}{3}t^2$ for set partitions and mean $\frac{4}{3}t^2$ for permutations, and with high probability all other components have size one or two. These approximations are proved, with preasymptotic error bounds, using combinatorial bijections for placements of $r$ rooks on a triangular half of an $n\times n$ chess board, together with the Chen--Stein method for processes of indicator random variables.