On the number of transversals in latin squares

The logarithm of the maximum number of transversals over all latin squares of order $n$ is greater than $\frac{n}{6}(\ln n+ O(1))$

On the number of transversals in a class of Latin squares

Denote by $\mathcal{A}_p^k$ the Latin square of order $n=p^k$ formed by the Cayley table of the additive group $(\mathbb{Z}_p^k,+)$, where $p$ is an odd prime and $k$ is a positive integer. It is shown that for each $p$ there exists $Q>0$ such that for all sufficiently large $k$, the number of transversals in $\mathcal{A}_p^k$ exceeds $(nQ)^{\frac{n}{p(p-1)}}$

Additive triples of bijections, or the toroidal semiqueens problem

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 $(\mathbb{Z}/n\mathbb{Z})^n$.Comment: 22 page