Cardinality of Rauzy classes

TL;DR

This paper derives an explicit formula for the number of elements in each Rauzy class, which are partitions of irreducible permutations used in interval exchange transformations.

Contribution

It provides the first explicit formula for the cardinality of Rauzy classes, advancing understanding of their structure and enumeration.

Findings

Explicit formula for Rauzy class cardinality

Enhanced understanding of permutation partitions in dynamical systems

Foundation for further combinatorial and dynamical analysis

Abstract

Rauzy classes define a partition of the set of irreducible (or indecomposable) permutations. They were defined by G. Rauzy as part of an induction algorithm for interval exchange transformations. In this article we prove an explicit formula for the cardinality of all Rauzy classes.