Attacks and alignments: rooks, set partitions, and permutations

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 similar to t root 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 2/3t(2) for set partitions and mean 4/3t(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 x n chess board, together with the Chen-Stein method for processes of indicator random variables.