Combinatorial bijections from hatted avoiding permutations in $S_n(132)$ to generalized Dyck and Motzkin paths

TL;DR

This paper introduces hatted permutation patterns, explores their growth rates, and establishes explicit bijections with Dyck and Motzkin paths, revealing new combinatorial correspondences and enumeration methods.

Contribution

It defines hatted patterns as a generalization of barred patterns and connects them to classical paths through explicit bijections, including a new bijection with Motzkin paths.

Findings

Hatted pattern avoiding permutations grow at a rate comparable to barred patterns.

Dyck paths with no peak at height p are counted by hatted pattern avoiding permutations.

A new bijection links Motzkin paths to permutations without consecutive adjacent numbers.

Abstract

We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to barred pattern. We prove that Dyck paths with no peak at height $p$, Dyck paths with no $ud... du$ and Motzkin paths are counted by hatted pattern avoiding permutations in $\s_n(132)$ by showing explicit bijections. As a result, a new direct bijection between Motzkin paths and permutations in $\s_n(132)$ without two consecutive adjacent numbers is given. These permutations are also represented on the Motzkin generating tree based on the Enumerative Combinatorial Object (ECO) method.