66 research outputs found
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
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
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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
Modular Decomposition and the Reconstruction Conjecture
We prove that a large family of graphs which are decomposable with respect to
the modular decomposition can be reconstructed from their collection of
vertex-deleted subgraphs.Comment: 9 pages, 2 figure
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
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
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