Printed in Great Britain Processes generating permutation expansions

This paper is concerned with processes by which a vector of n positive elements can be expanded as a sequence of permutations. The principal transformation considered is shown to generate a unique permutation expansion from every normalized vector, and the question of rate of convergence is examined. Other similar transformations are discussed less com-pletely. The idea of considering such transformations was suggested by an empirical investigation of M. G. Kendall. 1. Introduction. Kendall (1962) has described a process by which a set of n numbers is rearranged in ascending order, the ordered numbers are differenced, the n — 1 differences are rearranged in order, differences are again taken and so on. From information provided only by the rank orders of the differences at each stage, and with a final adjustment of location and scale, he has been able to regenerate the original numbers to a remarkable degree o