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.

A Note on Numerical Representations of Nested System of Strict Partial Orders

This note provides two numerical representations of a nested system of strict partial orders. The first representation is based on utility and threshold functions. We generalize the threshold representation of menu-dependent preferences by allowing the threshold to depend not only on the menu but also on the pair of alternatives under comparison. The threshold function can be interpreted as the distance between alternatives. The second representation is based on the aggregation of multi-dimensional preference. This representation describes a decision-making procedure where multiple criteria are gradually aggregated into an overall assessment.