Formulae for the Alon–Tarsi Conjecture

Abstract

The sign of a Latin square is $-1$ if it has an odd number of rows and columns that are odd permutations; otherwise it is $+1$. Let $L^{\text{\scshape{even}}}_n$ and $L^{\text{\scshape{odd}}}_n$ be, respectively, the number of Latin squares of order n with sign $+1$ and $-1$. The Alon–Tarsi conjecture asserts that $L^{\text{\scshape{even}}}_n \neq L^{\text{\scshape{odd}}}_n$ when n is even. We prove that $L_n^{\text{\scshape{even}}}-L_n^{\text{\scshape{odd}}}=(-1)^{n(n-1)/2} \sum_{A \in B_n} (-1)^{\sigma_0(A)} \det(A)^n$, where $B_n$ is the set of $n \times n$ $\,(0,1)$-matrices and $\sigma_0(A)$ is the number of 0 elements in A. We use this formula to give another proof of the Alon–Tarsi conjecture for $n=p-1$ for odd prime p.