Geodesic
Splittability and 1-amalgamability of permutation classes
Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2.

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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. 
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Jelínek, Vít; Opler, Michal. Splittability and 1-amalgamability of permutation classes. Discrete mathematics & theoretical computer science, Permutation Patterns 2016, Tome 19 (2017-2018) no. 2. doi : 10.23638/DMTCS-19-2-4. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-19-2-4/

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