2025/03/24 by Kumari, Nishu
#05B20 #05E05 #05E10 #15B35 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2503.18685
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 OSASMs and derived a product formula for counting the number of OSASMs of any fixed even order. In this work, we provide a product formula for the number of odd-order OSASMs of any fixed order. Additionally, we present an algebraic proof of a symmetry property for even-order OSASMs. This resolves all the three conjectures of Behrend, Fischer, and Koutschan (arXiv, 2023) regarding the exact enumeration of OSASMs.