The Cayley determinant of the determinant tensor and the Alon-Tarsi conjecture
Each row and column of a Latin square, \(L\), of order \(n\) can be thought of as a permutation of the elements in \(\{1,2,\dots,n\}\). \(L\) is even (odd) if the product of its row and column permutations is even (odd). A semi-normalized diagonal \(L\) is one for which the elements in the first column appear in natural order and only 1's appear on the diagonal. Let \(\text{els}(n)\), respectively \(\text{ols}(n)\), denote the number of even, respectively odd, Latin squares of order \(n\), and affix a prefix of ``snd'' if we restrict to semi-normalized diagonal \(L\)'s. If \(n\) is odd, then \(\text{els}(n)-\text{ols}(n)= 0\). If \(n\) is even the Alon-Tarsi conjecture states that this difference is nonzero. The author defines the Alon-Tarsi constant \(\text{AT}(n)\) which equals \(\text{sndels}(n)- \text{sndols}(n)\), and notes it is not necessarily \(0\) for odd \(n\). Indeed, the author extends the Alon-Tarsi conjecture to \(\text{AT}(n)\neq 0\) for every positive integer \(n\). The author shows that \(\text{els}(n)- \text{ols}(n)\) is nonnegative, and proves the Alon-Tarsi conjecture for Latin squares of order \(c2^r\), where \(r\in\mathbb{Z}^+\) and either \(c\) is an even integer for which the Alon-Tarsi conjecture is true, or \(c\) is an odd integer such that the extended Alon-Tarsi conjecture is true for both \(c\) and \(c+1\). Coupled with recent work of \textit{Arthur A. Drisko} [Adv. Math. 128, No. 1, 20-35 (1997; Zbl 0885.05034), above], this estabishes that the Alon-Tarsi conjecture is true for Latin squares of order \(2^r(p+1)\), where \(p\) is an odd prime.
scientific article; zbMATH DE number 1051590