On even and odd latin squares
Each row and column of a latin square, \(L\), of order \(n\) can be thought of as a permutation of the elements of an \(n\)-set. \(L\) is row-even (row- odd) if \(L\) has an even (odd) number of rows that are odd permutations. Column-even and column-odd are similarly defined. \(L\) is even if its rows and columns have the same parity; \(L\) is odd otherwise. The author examines these parities under the actions of the so-called isotopy and transformation groups. Let \(n\) be even. A shorter proof of the equivalence of the Alon-Tarsi (the number of even is not the number of odd latin squares of order \(n\)) and Huang-White (the number of row-even is not the number of row-odd latin squares of order \(n\)) conjectures is given. Also, it is shown that the number of row-even latin squares equals the number of even latin squares of order \(n\) if \(n\equiv 0\pmod 4\), and equals the number of odd latin squares if \(n\equiv 2\pmod 4\). With the aid of a computer, the Alon-Tarsi conjecture is established up through \(n= 8\).
scientific article; zbMATH DE number 720835