Additive triples of bijections, or the toroidal semiqueens problem

TL;DR

This paper establishes an asymptotic count for additive triples of bijections on cyclic groups, linking combinatorial, algebraic, and geometric problems through an analytic number theory approach.

Contribution

It introduces a novel application of the Hardy--Littlewood circle method to count additive triples of bijections in cyclic groups, connecting multiple combinatorial problems.

Findings

Asymptotic formula for the number of additive triples of bijections

Equivalence between counting additive triples, orthomorphisms, and nonattacking semiqueens

Method adapts the circle method to the group $(\mathbb{Z}/n\mathbb{Z})^n$

Abstract

We prove an asymptotic for the number of additive triples of bijections $\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs of bijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ 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 $(\mathbb{Z}/n\mathbb{Z})^n$.