Permutations with restricted patterns and Dyck paths

TL;DR

This paper establishes bijections between pattern-avoiding permutations and Dyck paths, enabling the derivation of generating functions and asymptotic estimates for permutations with specific pattern occurrences.

Contribution

It introduces new bijections linking 132- and 123-avoiding permutations to Dyck paths, connecting permutation enumeration to Motzkin path results and continued fractions.

Findings

Generated explicit formulas for pattern-avoiding permutations

Derived asymptotic estimates for permutations with fixed pattern counts

Connected permutation enumeration to classical path enumeration results

Abstract

We exhibit a bijection between 132-avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of the pattern $12... k$ follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet. As a bonus, we use these observations to derive further results and a precise asymptotic estimate for the number of 132-avoiding permutations of $\{1,2,...,n\}$ with exactly $r$ occurrences of the pattern $12... k$. Second, we exhibit a bijection between 123-avoiding permutations and Dyck paths. When combined with a result of Roblet and Viennot, this bijection allows us to express the generating function for 123-avoiding permutations with a given number of occurrences of the pattern $(k-1)(k-2)... 1k$ in form of a continued fraction and to derive further results for these permutations.