  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.
