On extensions of the Alon-Tarsi Latin Square conjecture

TL;DR

This paper explores extensions of the Alon-Tarsi Latin square conjecture by establishing new identities and relations involving permanents and determinants of matrices, specifically focusing on odd dimensions and (0,1)-matrices.

Contribution

It introduces new identities and connections between permanents and determinants that extend the Alon-Tarsi conjecture to odd-dimensional cases.

Findings

Derived an identity involving an alternating sum of permanents of (0,1)-matrices.

Linked products of permanents and determinants to conjectures extending Alon-Tarsi.

Provided theoretical insights into the structure of Latin square conjectures for odd dimensions.

Abstract

Expressions involving the product of the permanent with the (n-1)th power of the determinant of a matrix of indeterminates, and of (0,1)-matrices, are shown to be related to two conjectures that extend the Alon-Tarsi Latin square conjecture to odd dimensions. An identity involving an alternating sum of permanents of (0,1)-matrices is obtained.