Totally Positive Completable Matrix Patterns and Expansion
David H. Allen · W&M Publish (College of William & Mary) · 2019
Though some special cases are now understood, the characterization of TP-completable patterns is far from complete. Here, a new idea is developed: the \underline{expansion} of a pattern. It is used to explain some recent results, such as border patterns. The effects of expansion on certain cases of non-completable and completable patterns is examined, as well as an attempt to characterize $3$-by-$n$ TP-completable patterns. While many TP-completable patterns remain so under expansion, a counterExample shows that this is not always so. In the process, some new results about TP-completability are given.