On the Cardinality of Permutations for Interval Exchange Transformations
Ting Kuo · 2014
The basic problem of ＂enumerative combinatorics＂ is counting the number of elements for a set. This paper focuses on a particular set G_N, which is the subset of permutations of ＂N＂ items for ＂interval exchange transformations.＂ In mathematics, an ＂interval exchange transformation＂ is a type of dynamical system. Unlike a ＂sieve＂ method that begins with a larger set and somehow eliminates the unqualified elements, a decomposition approach was used in this study. Based on the results of using this approach, we propose a concise formula of the cardinality of G_N. In addition, we related the set of G_(N,N) to the set of B_(N,N), where G_(N,N) denotes the subset of G_N that is composed of all permutations with a prefix ＂N＂, and B_(N,N) denotes the set of permutations without a succession. For N ≥ 1, we proved and thus propose that B_(N,N) and G_(N+1,N+1) are ＂isomorphic＂ and that B_(N,N) is ＂postequivalent＂ to G_(N+1,N+1).