Motzkin Intervals and Valid Hook Configurations

TL;DR

This paper introduces a new partial order on Motzkin paths, establishes bijections with valid hook configurations of pattern-avoiding permutations, and explores their enumeration and asymptotic behavior.

Contribution

It defines a novel partial order on Motzkin paths, connects it with valid hook configurations of pattern-avoiding permutations, and provides enumeration and conjectures linking these structures.

Findings

Bijection between valid hook configurations and intervals in the new posets.

Counting formulas for valid hook configurations avoiding certain patterns.

Asymptotic enumeration of these configurations.

Abstract

We define a new natural partial order on Motzkin paths that serves as an intermediate step between two previously-studied partial orders. We provide a bijection between valid hook configurations of $312$-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding $132$ (or equivalently, $231$) are counted by the same numbers that count intervals in the Motzkin-Tamari posets that Fang recently introduced, and we give an asymptotic formula for these numbers. We then proceed to enumerate valid hook configurations of permutations avoiding other collections of patterns. We also provide enumerative conjectures, one of which links valid hook configurations of $312$-avoiding permutations, intervals in the new posets we have defined, and certain closed lattice walks with small steps that are confined to a quarter plane.