333 research outputs found
The enumeration of three pattern classes using monotone grid classes
The structure of the three pattern classes defined by the sets of forbidden permutations \{2143,4321\}, \{2143,4312\} and \{1324,4312\} is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined
Grid classes and partial well order
We prove necessary and sufficient conditions on a family of (generalised)
gridding matrices to determine when the corresponding permutation classes are
partially well-ordered. One direction requires an application of Higman's
Theorem and relies on there being only finitely many simple permutations in the
only non-monotone cell of each component of the matrix. The other direction is
proved by a more general result that allows the construction of infinite
antichains in any grid class of a matrix whose graph has a component containing
two or more non-monotone-griddable cells. The construction uses a
generalisation of pin sequences to grid classes, together with a number of
symmetry operations on the rows and columns of a gridding.Comment: 22 pages, 7 figures. To appear in J. Comb. Theory Series
- …