1,468 research outputs found
On partial well-order for monotone grid classes of permutations
A monotone grid class is a permutation class (i.e., a downset of permutations
under the containment order) defined by local monotonicity conditions. We give
a simplified proof of a result of Murphy and Vatter that monotone grid classes
of forests are partially well-ordered
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
Small permutation classes
We establish a phase transition for permutation classes (downsets of
permutations under the permutation containment order): there is an algebraic
number , approximately 2.20557, for which there are only countably many
permutation classes of growth rate (Stanley-Wilf limit) less than but
uncountably many permutation classes of growth rate , answering a
question of Klazar. We go on to completely characterize the possible
sub- growth rates of permutation classes, answering a question of
Kaiser and Klazar. Central to our proofs are the concepts of generalized grid
classes (introduced herein), partial well-order, and atomicity (also known as
the joint embedding property)
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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2Ă2 monotone grid classes are finitely based
In this note, we prove that all 2Ă2 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 2Ă2 (generalized) grid classes having two monotone cells in the same row
Inflations of geometric grid classes of permutations
All three authors were partially supported by EPSRC via the grant EP/J006440/1.Geometric grid classes and the substitution decomposition have both been shown to be fundamental in the understanding of the structure of permutation classes. In particular, these are the two main tools in the recent classification of permutation classes of growth rate less than Îș â 2.20557 (a specific algebraic integer at which infinite antichains first appear). Using language- and order-theoretic methods, we prove that the substitution closures of geometric grid classes are well partially ordered, finitely based, and that all their subclasses have algebraic generating functions. We go on to show that the inflation of a geometric grid class by a strongly rational class is well partially ordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less than Îș has a rational generating function. This bound is tight as there are permutation classes with growth rate Îș which have nonrational generating functions.PostprintPeer reviewe
On the effective and automatic enumeration of polynomial permutation classes
We describe an algorithm, implemented in Python, which can enumerate any
permutation class with polynomial enumeration from a structural description of
the class. In particular, this allows us to find formulas for the number of
permutations of length n which can be obtained by a finite number of block
sorting operations (e.g., reversals, block transpositions, cut-and-paste
moves)
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