Patterns of Alternating Sign Matrices

Authors:
  • RBRichard A. Brualdi
    Richard A. Brualdi
  • KKKathleen P. Kiernan
    Kathleen P. Kiernan
  • SMSeth A. Meyer
    Seth A. Meyer
  • MSMichael W. Schroeder
    Michael W. Schroeder
Preprint
Publisher website
Google Scholar
Abstract
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign matrices whose patterns are maximal within the class of all n by n alternating sign matrices.