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.Comment: Submitted to Advances in Applied Mathematic

Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. As we demonstrate, the numerical data for these problems demonstrates some surprising behavior. This strange behavior even provides some very unexpected data related to the number of 1324-avoiding permutations

Inflations of Geometric Grid Classes: Three Case Studies

We enumerate three specific permutation classes defined by two forbidden patterns of length four. The techniques involve inflations of geometric grid classes

Generating Permutations with Restricted Containers

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions