Restricted permutations

Restricted permutations are those constrained by having to avoid subsequences ordered in various prescribed ways. They have functioned as a convenient descriptor for several sets of permutations which arise naturally in combinatorics and computer science. We study the partial order on permutations (and more general sequences) that underlies the idea of restriction and which gives rise to sets of sequences closed under taking subsequences. In applications, the question of whether a closed set has a finite basis is often considered. We give a family of sets that have a finite basis, and then use them to study the inverse problem of describing a closed set from its basis. We give enumeration results in all cases where the basis consists of a permutationof length 3 and a permutation of length 4. The paper is an extended version of [2] in that it contains full proofs of the results presented there which have also been obtained by West [15].