On cycles in monotone grid classes of permutations

TL;DR

The paper develops a structural framework for monotone grid classes by introducing the $M$-sum and coil concepts to decompose grid permutations according to the cycle structure of the row–column graph $G_M$. It proves a precise characterisation of labelled well quasi-ordering for subclasses of pseudoforest grid classes, establishes decidability results, and shows that unicyclic grid classes are finitely based while producing pseudoforest examples not finitely based, thereby answering longstanding conjectures. The results illuminate how coil decompositions control the presence of infinite antichains and enable reductions to acyclic grids, yielding a nuanced picture of wqo, lwqo, and base finiteness across cyclic, unicyclic, and pseudoforest classes. Collectively, the work sharpens our understanding of the interaction between grid structure, decomposition, and order-theoretic properties with implications for enumeration and algorithmic decidability.

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.

Figures (15)

• Figure 1: The permutation $812543697$ possesses six griddings in $\mathsf{Grid}(M)$ where $M={}$, three of which are shown here.
• Figure 2: A gridding of the permutation $10\,1\,12\,2\,5\,4\,6\,3\,7\,11\,9\,8$ in $\mathsf{Grid}(M)$ where $M={}$is a partial multiplication matrix with column sequence $1,-1,-1$ and row sequence $1,-1,1$. The inherited orientations for each non-empty cell are shown in grey.
• Figure 3: The piecewise maps $f_{ij}$, described in the proof of Proposition \ref{['prop-doubling-equal']}, for a gridding of the permutation 2 11 14 16 3 6 4 7 1 8 12 5 9 13 10 15 in $\mathsf{Grid}(M)$, where $M={}$.
• Figure 4: A gridding of the permutation $415362$ in $\mathsf{Grid}(M)$ where $M={}$is a partial multiplication matrix, together with its orientation digraph.
• Figure 5: For $M={}$, the decomposition into $M$-indivisibles of the $M$-gridded permutation $\pi^\# = 2\,16\,14\,11\,3\,6\,4\,7\,1\,8\,12\,5\,9\,13\,10\,15$.

Theorems & Definitions (61)

• Proposition 2.1: Vatter and Waton vatter:on-partial-well:
• Proposition 2.2
• proof
• Proposition 2.3: Waton waton:on-permutation-:
• proof
• Proposition 2.4
• proof
• Definition 3.1: $M$-sum
• Lemma 3.3
• Lemma 3.5