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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 asimple approximation for the entire process of component counts; in particular,the number of components of size 3 converges in distribution to Poisson withmean $\frac{2}{3}t^2$ for set partitions and mean $\frac{4}{3}t^2$ forpermutations, and with high probability all other components have size one ortwo. These approximations are proved, with preasymptotic error bounds, usingcombinatorial 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 ofindicator random variables.

Attacks and alignments: rooks, set partitions, and permutations