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Counting and Generating Permutations Using Timed Languages

  • Nicolas Basset
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

The signature of a permutation σ is a word sg(σ) ⊆ {a, d}* whose i th letter is d when σ has a descent (i.e. σ(i) > σ(i + 1)) and is a when σ has an ascent (i.e. σ(i) < σ(i + 1)). Combinatorics of permutations with a prescribed signature is quite well explored. Here we state and address the two problems of counting and randomly generating in the set sg − 1(L) of permutations with signature in a given regular language L ⊆ {a, d}*. First we give an algorithm that computes a closed form formula for the exponential generating function of sg − 1(L). Then we give an algorithm that generates randomly the n-length permutations of sg − 1(L) in a uniform manner, that is all the permutations of a given length with signature in L are equally probable to be returned. Both contributions are based on a geometric interpretation of a subclass of regular timed languages.

Keywords

Regular Language Closed Form Formula Clock Word Generate Permutation Exponential Generate Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Nicolas Basset
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordUnited Kingdom

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