THE CAYLEY DETERMINANT OF THE DETERMINANT TENSOR AND THE ALON-TARSI CONJECTURE

Given a base on a vector space of dimension n, we can represent a tens or of order r with a hypermatrix of dimension n and order r. Then, the standard determinant tensor is represented by a hypermatrix H of orde r and dimension n. Gherardelli showed that the Cayley determinant of H , times n!, is equal to the number of even Latin squares of order n mi nus the number of odd Latin squares of order n. The Alon-Tarsi conject ure says that this difference is not zero, whenever n is even. If n is odd the difference is zero, but the conjecture can be extended to the odd case by computing the difference only for Latin squares which hav e the entries of the diagonal equal to 1. In this paper we use the Lap lace rule in order to compute the Cayley determinant, and we prove tha t the difference between the number of even Latin squares and the numb er of odd Latin squares is nonnegative. We also prove the Alon-Tarsi c onjecture for Latin squares of order c2(r), where r is a positive inte ger and either c is an even integer for which the Alon-Tarsi conjectur e is true, or c is an odd integer such that the extended Alon-Tarsi co njecture is true for c and for c + 1. (C) 1997 Academic Press.