2×2 monotone grid classes are finitely based

In this note, we prove that all 2×2 monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain 2×2 (generalized) grid classes having two monotone cells in the same row

2×22\times 2 monotone grid classes are finitely based

In this note, we prove that all $2 \times 2$ monotone grid classes are finitely based, i.e., defined by a finite collection of minimal forbidden permutations. This follows from a slightly more general result about certain $2 \times 2$ (generalized) grid classes having two monotone cells in the same row.Comment: 10 pages, 5 figures. To appear in Discrete Mathematics and Theoretical Computer Science, special issue for Permutation Patterns 201

Subclasses of the separable permutations

We prove that all subclasses of the separable permutations not containing Av(231) or a symmetry of this class have rational generating functions. Our principal tools are partial well-order, atomicity, and the theory of strongly rational permutation classes introduced here for the first time

Wreath Products of Permutation Classes

A permutation class which is closed under pattern involvement may be described in terms of its basis. The wreath product construction X \wr Y of two permutation classes X and Y is also closed, and we investigate classes Y with the property that, for any finitely based class X, the wreath product X \wr Y is also finitely based.Comment: 14 page

Permutation Classes of Polynomial Growth

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and sufficient conditions on sets of forbidden permutations which ensure that the associated pattern class is of polynomial growth is determined. A catalogue of all such sets of forbidden permutations having three or fewer elements is provided together with bounds on the degrees of the associated enumerating polynomials.Comment: 17 pages, 4 figure

Pattern avoidance classes and subpermutations

Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function.Comment: 18 pages, 4 figures (all in-line