There are asymptotically the same number of Latin squares of each parity
  A Latin square is reduced if its first row and column are in natural order. For Latin squares of a particular order $n$ there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow\infty$. 
Nicholas J. Cavenagh and Ian M. Wanless
math.CO