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Geometric grid classes and the substitution decomposition have both beenshown to be fundamental in the understanding of the structure of permutationclasses. In particular, these are the two main tools in the recentclassification of permutation classes of growth rate less than$\kappa\approx2.20557$ (a specific algebraic integer at which infiniteantichains begin to appear). Using language- and order-theoretic methods, weprove that the substitution closures of geometric grid classes are partiallywell-ordered, finitely based, and that all their subclasses have algebraicgenerating functions. We go on to show that the inflation of a geometric gridclass by a strongly rational class is partially well-ordered, and that all itssubclasses have rational generating functions. This latter fact allows us toconclude that every permutation class with growth rate less than $\kappa$ has arational generating function. This bound is tight as there are permutationclasses with growth rate $\kappa$ which have nonrational generating functions.

Inflations of geometric grid classes of permutations