Additive triples of bijections, or the toroidal semiqueens problem (Q1725577)

Summary: We prove an asymptotic for the number of additive triples of bijections \(\{1,\dots,n\}\rightarrow\mathbb{Z}/n\mathbb{Z}\), that is, the number of pairs of bijections \(\pi_1,\pi_2\colon \{1,\dots,n\}\rightarrow\mathbb{Z}/n\mathbb{Z}\) such that the pointwise sum \(\pi_1+\pi_2\) is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of \(\mathbb{Z}/n\mathbb{Z}\), to counting the number of arrangements of \(n\) mutually nonattacking semiqueens on an \(n\times n\) toroidal chessboard, and to counting the number of transversals in a cyclic Latin square. The method of proof is a version of the Hardy-Littlewood circle method from analytic number theory, adapted to the group \((\mathbb{Z}/n\mathbb{Z})^n\).