Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding

TL;DR

This paper proves that permutation classes avoiding certain bipartite partially ordered patterns have regular insertion encodings and rational generating functions, and it provides combinatorial specifications for many classes, resolving several conjectures.

Contribution

It establishes a general result on the regularity of insertion encodings for classes avoiding POPs of height at most two and extends this to many other classes using combinatorial exploration.

Findings

Permutation classes avoiding height ≤ 2 POPs have rational generating functions.

Combinatorial exploration yields specifications for hundreds of classes.

Several conjectures by Gao, Kitaev, Chen, and Lin are resolved.

Abstract

We prove that any class of permutations defined by avoiding a partially ordered pattern (POP) with height at most two has a regular insertion encoding and thus has a rational generating function. Then, we use Combinatorial Exploration to find combinatorial specifications and generating functions for hundreds of other permutation classes defined by avoiding a size 5 POP, allowing us to resolve several conjectures of Gao and Kitaev and of Chen and Lin.