Elizalde, Sergi - Continued fractions for permutation statistics

dmtcs:4602 - Discrete Mathematics & Theoretical Computer Science, June 25, 2018, Vol. 19 no. 2, Permutation Patterns 2016
Continued fractions for permutation statistics

Authors: Elizalde, Sergi

We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle diagrams, we find simple translations of some statistics on permutations (and subsets of permutations) into statistics on colored Motzkin paths, which are amenable to the use of continued fractions. We obtain new enumeration formulas for subsets of permutations with respect to fixed points, excedances, double excedances, cycles, and inversions. In particular, we prove that cyclic permutations whose excedances are increasing are counted by the Bell numbers.

Source : oai:arXiv.org:1703.08742
Volume: Vol. 19 no. 2, Permutation Patterns 2016
Section: Permutation Patterns
Published on: June 25, 2018
Submitted on: August 1, 2017
Keywords: Mathematics - Combinatorics,05A05 (Primary) 05A15, 05A19, 30B70, 05A18 (Secondary)