The Cayley Determinant of the Determinant Tensor and the Alon–Tarsi Conjecture

Abstract

Given a base on a vector space of dimensionn, we can represent a tensor of orderrwith a hypermatrix of dimensionnand orderr. Then, the standard determinant tensor is represented by a hypermatrixHof order and dimensionn. Gherardelli showed that the Cayley determinant ofH, timesn!, is equal to the number of even Latin squares of ordernminus the number of odd Latin squares of ordern. The Alon–Tarsi conjecture says that this difference is not zero, whenevernis even. Ifnis odd the difference is zero, but the conjecture can be extended to the odd case by computing the difference only for Latin squares which have the entries of the diagonal equal to 1. In this paper we use the Laplace rule in order to compute the Cayley determinant, and we prove that the difference between the number of even Latin squares and the number of odd Latin squares is nonnegative. We also prove the Alon–Tarsi conjecture for Latin squares of orderc2r, whereris a positive integer and eithercis an even integer for which the Alon-Tarsi conjecture is true, orcis an odd integer such that the extended Alon-Tarsi conjecture is true forcand forc+1.