Abstract
The paper links four conjectures: (1) (Rota's bases conjecture): For any system A = ((1)A, . . . , (n)A) of non-singular real valued matrices the multiset of all columns of matrices in A can be decomposed into n independent systems of representatives of A. (2) (Alon-Tarsi): For even n, the number of even n x n Latin squares differs from the number of odd n x n Latin squares. (3) (Stones-Wanless, Kotler): For all n, the number of even n x n Latin squares with the identity permutation as first row and first column differs from the number of odd n x n Latin squares of this type. (4) (Aharoni-Berger): Let M and N be two matroids on the same vertex set, and let A(1), . . . , A(n) be sets of size n+1 belonging to the intersection of M and N. Then there exists a set belonging to the intersection of M and N meeting all A(i).
Huang and Rota [8] and independently Onn [11] proved that for any n (2) implies (1). We prove equivalence between (2) and (3).
Using this, and a special case of (4), we prove the Huang-Rota-Onn theorem for n odd and a restricted class of input matrices: assuming the Alon-Tarsi conjecture for n-1, Rota's conjecture is true for any system of non-singular real valued matrices where one of them is non-negative and the remaining have non-negative inverses.