Systems of distinct representatives and linear algebra

Abstract

Some purposes of this paper are: (1) To take seriously the term , 'term rank.' (2) To make an issue of not 'rearranging rows and columns' by not 'arranging' them in the first place. (3) To promote the numerical use of Cramer's rule. (4) To illustrate that the relevance of 'number of steps' to 'amount of work' depends on the amount of work in a step. (5) To call attention to the computational aspect of SDR's, an aspect where the subject differs from being an instance of familiar linear algebra. (6) To describe an SDR instance of a theory on extremal combinatorics that uses linear algebra in very different ways than does totally unimodular theory. (The preceding paper, Optimum Branchings, describes another instance of that theory.)