Totally positive shapes and TP k-completable patterns

Abstract

The notions of total positivity and of TP k are generalized to "shapes" (a generalization of matrices). In particular, the relationship between positivity of "contiguous" minors and all minors is characterized for general shapes and for certain special types of shapes. This and other ideas are used to address the TP k-completion problem and TP k-completable patterns. In case k=2, a near characterization of TP 2-completable patterns is given. © 2011 Elsevier Inc. All rights reserved.