Inflations of geometric grid classes of permutations
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Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than κ ≈ 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than κ has a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.
Albert , M D , Ruskuc , N & Vatter , V 2015 , ' Inflations of geometric grid classes of permutations ' , Israel Journal of Mathematics , vol. 205 , no. 1 , pp. 73-108 . https://doi.org/10.1007/s11856-014-1098-8
Israel Journal of Mathematics
© 2014. The Hebrew University Magnes Press. This is the accepted version of the following article: This is the preprint version before acceptance of the following article: Exact dimensionality and projections of random self-similar measures and sets Falconer, K. & Jin, X. 2014 In : Journal of the London Mathematical Society. The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-014-1098-8