On cycles in monotone grid classes of permutations

TL;DR

This paper introduces a new decomposition called the $M$-sum to analyze permutations in monotone grid classes with specific graph properties, leading to characterizations of well quasi-ordering and finite basis status, and disproving a prior conjecture.

Contribution

It develops the $M$-sum decomposition, characterizes well quasi-ordering in certain grid classes, and provides counterexamples to a 2006 conjecture.

Findings

Characterization of when subclasses are well quasi-ordered.

Identification of conditions for finite basis in grid classes.

Counterexamples disproving the 2006 conjecture.

Abstract

We undertake a detailed investigation into the structure of permutations in monotone grid classes whose row-column graphs do not contain components with more than one cycle. Central to this investigation is a new decomposition, called the $M$-sum, which generalises the well-known notions of direct sum and skew sum, and enables a deeper understanding of the structure of permutations in these grid classes. Permutations which are indecomposable with respect to the $M$-sum play a crucial role in the structure of a grid class and of its subclasses, and this leads us to identify coils, a certain kind of permutation which corresponds to repeatedly traversing a chosen cycle in a particular manner. Harnessing this analysis, we give a precise characterisation for when a subclass of such a grid class is labelled well quasi-ordered, and we extend this to characterise (unlabelled) well quasi-ordering in certain cases. We prove that a large general family of these grid classes are finitely based, but we also exhibit other examples that are not, thereby disproving a conjecture from 2006 due to Huczynska and Vatter.