Problems and Conjectures Presented at the Third International Conference on Permutation Patterns (university of Florida, March 7–11, 2005) Recounted by Murray Elder and Vince Vatter

A permutation is an arrangement of a finite number of distinct elements of a linear order, for example, e, π, 0, √ 2 and 3412. Two permutations are order isomorphic if the have the same relative ordering. We say a permutation τ contains or involves the permutation β if deleting some of the entries of τ gives a permutation that is order isomorphic to β, and we write β ≤ τ . For example, 534162 (when permutations contain only single digit natural numbers we suppress the commas) contains 321 (delete the values 4, 6, and 2). A permutation avoids a permutation if it does not contain it. For a set of permutations B define Av(B) to be the set of permutations that avoid all of the permutations in B and let sn(B) denote the number of permutations of length n in Av(B). A set of permutations or class C is closed if π ∈ C and σ ≤ π implies σ ∈ C. Therefore Av(B) is a class for every set of permutations B and every permutation class can be written as Av(B) for some set B. An antichain of permutations is a set of permutations such that no permutation contains another. We call a set of permutations B the basis of a class of permutations C if C = Av(B) and B is an antichain.