On the intriguing search for good permutations

Abstract

The intriguing search for permutations that generate generalised van der Corput sequences with exceptionally small discrepancy forms an important part of the research work of Henri Faure. The aim of these notes is to survey (some of) Henri's contributions over the last four decades which considerably improved our understanding of one-dimensional van der Corput sequences and inspired a lot of related work. We recall and compare the different approaches in the search for generalised van der Corput sequences with low discrepancy, i.e., using a single generating permutation versus using a sequence of permutations. Throughout, we collect and (in some cases) sharpen open questions which all stem from the extensive work of Henri and his coworkers and which will hopefully inspire more work in the future.