On the Stanley–Wilf limit of 4231-avoiding permutations and a conjecture of Arratia

We show that the Stanley–Wilf limit for the class of 4231-avoiding permutations is at least by 9.47. 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 Stanley–Wilf limit of a class of permutations avoiding a single permutation of length k cannot exceed (k−1)2. The result is established by constructing a sequence of finite automata that accept subclasses of the class of 4231-avoiding permutations and analysing their transition matrices.