We consider random set partitions of size $n$ with exactly $k$ blocks, chosen uniformly from all such, as counted by $S(n,k)$, the Stirling number of the second kind, and random permutations of size $n$ with exactly $k$ cycles, chosen uniformly from all such, as counted by $s(n,k)$, the unsigned Stirling number of the first kind, under the regime where $r \equiv r(n,k) := n-k \sim t\sqrt{n}$. 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 quantitative 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. 查看全文>>