Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order 2 mod 4, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving E₁,...,E_n, where E_i is the number of diagonals of a given Latin square that contain exactly i different symbols.

Let A(i|j) denote the matrix obtained by deleting row i and column j from a parent matrix A. Define t_ij to be the number of transversals in L(i|j), for some fixed Latin square L. We show that t_ab= t_cd mod 2 for all a,b,c,d and L. Also, if L has odd order then the number of transversals of L equals t_ab mod 2. We conjecture that t_ac + t_bc + t_ad + t_bd = 0 mod 4 for all a,b,c,d.

In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a k-regular bipartite graph on 2n vertices is divisible by 4 when n is odd and k=0 mod 4. We also show that per A(a|c)+per A(b|c)+per A(a|d)+per A(b|d) = 0 mod 4 for all a,b,c,d, when A is an integer matrix of odd order with all row and columns sums equal to k=2 mod 4.