Chapter 3 - Coalescence of Rows

Summary

INTRODUCTION

Coalescence of two rows of a matrix has been implicitly used by Ferguson [37, p. 14]. It has been formally defined by Karlin and Karon [70], who studied its influence on the regularity of the matrix and gave the Taylor expansion (3.5.2). Lorentz and Zeller [108] proved that the leading coefficient in this formula is different from 0. Lorentz [94] studied the coalescence of several rows. There are many applications of this method, for example, in [70, 91, 94, 103]. We discuss them in Chapter 4.

Let E = [e_{i, k}] be an m × (n + 1) matrix, not necessarily normal, satisfying the Pólya condition (1.4.3). We interpret E as a vertical grid of boxes. If e_{i, k} = 1, then a ball occupies the ith box in the kth column. We place a tray of n + 1 boxes under the column of the grid. Then the balls are permitted to fall from the grid into the boxes of the tray in such a way that if the box immediately below is occupied, the ball rolls to the first available box on the right. The condition (1.4.3) assures us that no ball will roll out of the tray. The distribution of balls in the tray constitutes the one-row matrix obtained by coalescence of the m rows of E. It is to be expected that the final arrangement of the balls in the tray is independent of the manner in which the balls were allowed to fall. Figure 3.1 is an example of coalescence of a two-row matrix.