On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture   of Arratia
  We construct a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations. We thereby show that the Wilf-Stanley limit for the class of 4231-avoiding permutations is bounded below by 9.35. This bound shows that this class has the largest such limit among all classes of permutations avoiding a single permutation of length 4 and refutes the conjecture that the Wilf-Stanley limit of a class of permutations avoiding a single permutation of length k cannot exceed (k-1)^2. 
M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott and M. Zabrocki