Georgia Institute of Technology College of Sciences

(Alon-Tarsi Conjecture): For even n, the number of even nxn Latin squares differs from the number of odd nxn Latin squares. (Stones-Wanless, Kotlar Conjecture): For all n, the number of even nxn Latin squares with the identity permutation as first row and first column differs from the number of odd nxn Latin squares of this type. (Aharoni-Berger Conjecture): Let M and N be two matroids on the same vertex set, and let A1,...,An be sets of size n + 1 belonging to both M and N. Then there exists a set belonging to both M and N and meeting all Ai. We prove equivalence of the first two conjectures and a special case of the third one and use these results to show that Alon-Tarsi Conjecture implies Rota's bases conjecture for odd n and any system of n non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.Joint work with Ron Aharoni.