Growth rates of permutation classes: categorization up to the uncountability threshold

Growth rates of permutation classes: from countable to uncountable

“…After establishing several delicate results about sum indecomposable permutations in a subsequent paper with Pantone [55], we are able to provide a complete characterization of growth rates under ξ, leading to the number line shown in Figure 1 of the introduction. Together with Bevan's Theorem 1.3, which shows that every real number greater than λ B « 2.36 is the growth rate of a permutation class, this leaves us tantalizingly close to the complete characterization of growth rates of permutation classes-the gap between the two results is approximately 0.05176.…”

“…In order to extend the approach used here and in [55] to complete this characterization, it seems that one would first want to establish the following. Before moving on, a few notes regarding Conjecture 10.1 are in order.…”

“…In Section 9 we show that all growth rates below ξ are growth rates of sum closed classes. Thus our characterization of these growth rates in [55] reduces to a characterization of the possible sequences of sum indecomposable classes, though that effort is far from trivial. It is not a coincidence that ξ is the growth rate of a sum closed class whose sum indecomposable members have the enumeration 1, 1, 2, 3, 4 8 (where 4 8 denotes an infinite sequence of 4s), though it is perhaps a coincidence that the same growth rate is achieved by a sum closed class whose sum indecomposable members have the enumeration 1, 1, 2, 4, 3, 3, 2, 1.…”

“…The proof of Theorem 9.7 is completed in Section 9, after we have built up a collection of tools to analyze the classes of interest. In [55] we establish the complete list of growth rates between κ and ξ, and in doing so establish that each of these growth rates is achieved by a finitely based class, whereas there are growth rates arbitrarily close to ξ which cannot be achieved by finitely based classes (because there are uncountably many such growth rates). This characterization and Bevan's Theorem 1.3 leave us tantalizingly close to the complete determination of the set of possible growth rates of permutation classes, as shown in Figure 1.…”

“…Bevan [22] V [62] Figure 1: The set of all growth rates of permutation classes between 2 and 2.5, as it stands after the characterization in Pantone and Vatter [55].…”