Abstract
Abstract The existence of a small partition of a combinatorial structure into random-like subparts, a so-called regular partition , has proven to be very useful in the study of extremal problems, and has deep algorithmic consequences. The main result in this direction is the renowned Szemeredi Regularity Lemma [Szemeredi, E., Regular partitions of graphs , Colloques Internationaux C.N.R.S. No: 260 – Problemes Combinatoires et Theorie des Graphes, Orsay (1976), 399–401] in graph theory. In this note, we are concerned with regularity in permutations: we show that every permutation of a sufficiently large set has a regular partition into a small number of intervals. This refines the partitioning given by Cooper [Cooper, J., A permutation regularity lemma , The Electronic Journal of Combinatorics 13 (2006), 20pp.], which required an additional non-interval exceptional class. We also introduce a distance between permutations that plays an important role in the study of convergence of a permutation sequence.
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