Grid classes and partial well order

TL;DR

This paper characterizes when permutation classes defined by grid matrices are partially well-ordered, using Higman's Theorem and constructions of infinite antichains based on the structure of the grid.

Contribution

It provides necessary and sufficient conditions for partial well-order in grid classes, extending previous results with new constructions and theoretical insights.

Findings

Conditions for partial well-order based on simple permutations

Construction of infinite antichains in certain grid classes

Application of Higman's Theorem to permutation classes

Abstract

We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and relies on there being only finitely many simple permutations in the only non-monotone cell of each component of the matrix. The other direction is proved by a more general result that allows the construction of infinite antichains in any grid class of a matrix whose graph has a component containing two or more non-monotone-griddable cells. The construction uses a generalisation of pin sequences to grid classes, together with a number of symmetry operations on the rows and columns of a gridding.