Problems and Conjectures Presented at the Fifth International Conference on Permutation Patterns (university of St Andrews, June 11–15, 2007) Recounted by Vince Vatter

We say a permutation π contains or involves the permutation σ if deleting some of the entries of π gives a permutation that is order isomorphic to σ, and we write σ ¤ π. For example, 534162 contains 321 (delete the values 4, 6, and 2). A permutation avoids a permutation if it does not contain it. This notion of containment defines a partial order on the set of all finite permutations, and the downsets of this order are called permutation classes. For a set of permutations B define AvpBq to be the set of permutations that avoid all of the permutations in B. Clearly AvpBq is a permutation class for every set B, and conversely, every permutation class can be expressed in the form AvpBq. For the problems we need one more bit of notation. Given permutations π and σ of lengths m and n, respectively, their direct sum, π`σ, is the permutation of length m n in which the first m entries are equal to π and the last n entries are order isomorphic to σ while their skew sum, πa σ, is the permutation of length m n in which the first m entries are order isomorphic to π while the last n entries are equal to π. For example, 231 ` 321 231654 and 231 a 321 564321.