### On Extensions of the Alon-Tarsi Latin Square Conjecture

#### Abstract

Expressions involving the product of the permanent with the $(n-1)^{st}$ power of the determinant of a matrix of indeterminates, and of (0,1)-matrices, are shown to be related to an extension to odd dimensions of the Alon-Tarsi Latin Square Conjecture, first stated by Zappa. These yield an alternative proof of a theorem of Drisko, stating that the extended conjecture holds for Latin squares of odd prime order. An identity involving an alternating sum of permanents of (0,1)-matrices is obtained.

#### Keywords

Latin square; Alon-Tarsi Latin Square conjecture; Parity of a Latin square; adjacency matrix; permanent of (0,1)-matrix