A permutation class is a set of permutations closed under taking permutation patterns. Thus, if a permutation belongs to the class, every permutation pattern it contains also belongs to the class. The class is proper if it is not the class of all finite permutations. The basis is the antichain of minimal excluded permutations; a class is finitely based when this basis is finite (Pantone and Vatter 2026).
If 
is the set of length-
members, its lower and upper growth rates are 
and 
. A proper growth rate exists
when these two finite quantities are equal. Pantone and Vatter (2026) constructed
a proper, infinitely based permutation class for which these rates differ. Whether
every proper finitely based permutation class has a growth rate remains open. Their
example does not contradict the Stanley-Wilf
conjecture, which concerns avoidance of a single permutation
pattern. Pantone and Vatter (2026) credit ChatGPT with an initial counterexample
argument that they verified and substantially simplified.
