Pattern classes of permutations via bijections between linearly ordered sets

A pattern class is a set of permutations closed under pattern involvement or, equivalently, defined by certain subsequence avoidance conditions. Any pattern class X which is atomic, i.e. indecomposable as a union of proper subclasses, has a representation as the set of subpermutations of a bijection between two countable (or finite) linearly ordered sets A and B . Concentrating on the situation where A is arbitrary and B = N , we demonstrate how the order-theoretic properties of A determine the structure of X and we establish results about independence, contiguity and subrepresentations for classes admitting multiple representations of this form.