Abstract
A permutation class is splittable if it is contained in the merge of two of its proper subclasses. We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases.
Highlights
In recent years one of the main areas of investigation within the study of pattern-avoiding permutations, or permutation classes has been to develop structural characterisations for some classes in terms of simpler ones
We characterise the unsplittable subclasses of the class of separable permutations both structurally and in terms of their bases
The electronic journal of combinatorics 23(2) (2016), #P2.49 monotone increasing or monotone decreasing permutations were all shown to be finitely based in [12] and played a role in [1]
Summary
In recent years one of the main areas of investigation within the study of pattern-avoiding permutations, or permutation classes has been to develop structural characterisations for some classes in terms of simpler ones. We delay formal definitions to the section but briefly a permutation π is a merge of two permutations σ and τ if its elements can be partitioned into two sets which are isomorphic (in the sense of relative ordering of corresponding pairs of points) to σ and τ respectively. This construction has been relatively ignored in the literature. For any permutation class C we define the basis of C, Ba(C), to be the set of minimal elements of S \ C (with respect to ).
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