Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes

The juxtaposition of permutation classes C and D is the class of all permutations formed by concatenations στ, such that σ is order isomorphic to a permutation in C, and τ to a permutation in D. We give simple necessary and sufficient conditions on the classes C and D for their juxtaposition to be labelled well-quasi-ordered (lwqo): namely that both C and D must themselves be lwqo, and at most one of C or D can contain arbitrarily long zigzag permutations. We also show that every class without long zigzag permutations has a growth rate which must be integral.