On the maximum number of Latin transversals

Let T(n)T(n) denote the maximal number of transversals in an order-n   Latin square. Improving on the bounds obtained by McKay et al., Taranenko recently proved that T(n)≤((1+o(1))ne2)n, and conjectured that this bound is tight.We prove via a probabilistic construction that indeed T(n)=((1+o(1))ne2)n. Until the present paper, no superexponential lower bound for T(n)T(n) was known. We also give a simpler proof of the upper bound.