A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.

Source : oai:arXiv.org:1704.08732

Volume: Vol. 19 no. 2, Permutation Patterns 2016

Section: Permutation Patterns

Published on: December 5, 2017

Submitted on: May 1, 2017

Keywords: Mathematics - Combinatorics,05A05

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