# Additive triples of bijections, or the toroidal semiqueens problem

### Sean Eberhard

London, UK### Freddie Manners

Stanford University, USA### Rudi Mrazović

University of Zagreb, Croatia

## Abstract

We prove an asymptotic for the number of additive triples of bijections ${1,…,n}→Z/nZ$, that is, the number of pairs of bijections $π_{1},π_{2}:{1,…,n}→Z/nZ$ such that the pointwise sum $π_{1}+π_{2}$ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of $Z/nZ$, to counting the number of arrangements of $n$ mutually nonattacking semiqueens on an $n×n$ 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 $(Z/nZ)_{n}$.

## Cite this article

Sean Eberhard, Freddie Manners, Rudi Mrazović, Additive triples of bijections, or the toroidal semiqueens problem. J. Eur. Math. Soc. 21 (2019), no. 2, pp. 441–463

DOI 10.4171/JEMS/841