Alternating sign and sign-restricted matrices: representations and partial orders

Sign-restricted matrices (SRMs) are $(0, \pm 1)$-matrices where, ignoring 0's, the signs in each column alternate beginning with a $+1$ and all partial row sums are nonnegative. The most investigated of these matrices are the alternating sign matrices (ASMs), where the rows also have the alternating sign property, and all row and column sums equal 1. We introduce monotone triangles to represent SRMs and investigate some of their properties and connections to certain polytopes. We also investigate two partial orders for ASMs related to their patterns alternating cycles and show a number of combinatorial properties of these orders.