Geometric grid classes of permutations

TL;DR

This paper studies geometric grid classes of permutations, showing they are finitely characterized, well-structured, and have rational generating functions, with these properties extending to their subclasses.

Contribution

It proves that geometric grid classes are finitely defined, partially well ordered, and have rational generating functions, and these properties are inherited by their subclasses.

Findings

Finite forbidden permutation sets characterize these classes.

They are partially well ordered.

Their generating functions are rational.

Abstract

A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope \pm1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.