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18
Geometric grid classes of permutations
 TRANS. AMER. MATH. SOC
, 2012
"... A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forb ..."
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Cited by 21 (10 self)
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A geometric grid class consists of those permutations that can be drawn on a specified set of line segments of slope ±1 arranged in a rectangular pattern governed by a matrix. Using a mixture of geometric and language theoretic methods, we prove that such classes are specified by finite sets of forbidden permutations, are partially well ordered, and have rational generating functions. Furthermore, we show that these properties are inherited by the subclasses (under permutation involvement) of such classes, and establish the basic lattice theoretic properties of the collection of all such subclasses.
Growing at a Perfect Speed
, 2007
"... A collection of permutation classes is exhibited whose growth rates form a perfect set, thereby refuting some conjectures of Balogh, Bollobás and Morris. ..."
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Cited by 10 (3 self)
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A collection of permutation classes is exhibited whose growth rates form a perfect set, thereby refuting some conjectures of Balogh, Bollobás and Morris.
Grid classes and partial well order
 J. Combin. Theory Ser. A
"... Abstract We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially wellordered. One direction requires an application of Higman's Theorem and relies on there being only finitely many simple ..."
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Cited by 7 (3 self)
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Abstract We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially wellordered. One direction requires an application of Higman's Theorem and relies on there being only finitely many simple permutations in the only nonmonotone 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 nonmonotonegriddable 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.
INFLATIONS OF GEOMETRIC GRID CLASSES OF PERMUTATIONS
"... 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 κ�2.20557 (a specifi ..."
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Cited by 6 (2 self)
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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 κ�2.20557 (a specific algebraic integer at which infinite antichains begin to appear). Using language and ordertheoretic methods, we prove that the substitution closures of geometric grid classes are partially wellordered, 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 partially wellordered, and that all its subclasses have rational generating functions. This latter fact allows us to conclude that every permutation class with growth rate less thanκhas a rational generating function. This bound is tight as there are permutation classes with growth rate κ which have nonrational generating functions.
The Approximability and Integrality Gap of Interval Stabbing and Independence Problems
"... Abstract Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of dintervals and dunionintervals. We obtain the following: (1) constructions yielding asymptotically tight lower bounds on the integrality g ..."
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Abstract Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of dintervals and dunionintervals. We obtain the following: (1) constructions yielding asymptotically tight lower bounds on the integrality gaps of the associated natural linear programming relaxations; (2) an LPrelative dapproximation for the hitting set problem on dintervals; (3) a proof that the approximation ratios for independent set on families of 2intervals and 2unionintervals can be improved to match tight duality gap lower bounds obtained via topological arguments, if one has access to an oracle for a PPADcomplete problem related to finding BorsukUlam fixedpoints.
WellQuasiOrder for Permutation Graphs Omitting a Path and a Clique
"... We consider wellquasiorder for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting P5 and a clique of any size is wellquasiordered. This is proved by giving a structural decomposition of the corresponding permut ..."
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We consider wellquasiorder for classes of permutation graphs which omit both a path and a clique. Our principle result is that the class of permutation graphs omitting P5 and a clique of any size is wellquasiordered. This is proved by giving a structural decomposition of the corresponding permutations. We also exhibit three infinite antichains to show that the classes of permutation graphs omitting {P6,K6}, {P7,K5}, and {P8,K4} are not wellquasiordered.