Counting and Generating Permutations Using Timed Languages

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.

This research is supported in part by ERC Advanced Grant VERIWARE and was also supported by the ANR project EQINOCS (ANR-11-BS02-004).

Omitted proofs and detailed examples can be found in Chapter 8 of [4].