Off-diagonally symmetric alternating sign matrices

A diagonally symmetric alternating sign matrix ( DSASM ) is a symmetric matrix with entries − 1 , 0 and 1, where the nonzero entries alternate in sign along each row and column, and the sum of the entries in each row and column equals 1. An off-diagonally symmetric alternating sign matrix ( OSASM ) is a DSASM , where the number of nonzero diagonal entries is 0 for even-order matrices and 1 for odd-order matrices. Kuperberg (Ann. Math., 2002) studied even-order OSASM s and derived a product formula for the number of OSASM s of any fixed even order. In this work, we provide a product formula for the number of OSASM s of any fixed odd order. Additionally, we present an algebraic proof of a symmetry property for even-order OSASM s. This resolves all three of the main conjectures of Behrend, Fischer, and Koutschan (arXiv, 2023) regarding the exact enumeration of OSASM s.