10.23638/DMTCS-19-2-11
Elizalde, Sergi
Sergi
Elizalde
Continued fractions for permutation statistics
Episciences.org
2018
Mathematics - Combinatorics
05A05 (Primary) 05A15, 05A19, 30B70, 05A18 (Secondary)
Article
https://dmtcs.episciences.org/4602
arXiv:1703.08742
1365-8050
PDF
Attribution-NoDerivs 2.0 Generic
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.
Discrete Mathematics & Theoretical Computer Science ; Vol. 19 no. 2, Permutation Patterns 2016 ; 1365-8050