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ABSTRACT: We prove that a hereditary property of cographs has bounded linear
clique-width if and only if it does not contain all quasi-threshold graphs or
their complements. The proof borrows ideas from the enumeration of permutation
classes, and the similarities between these two strands of investigation lead
us to a conjecture relating the graph properties of bounded linear clique-width
to permutation classes with rational generating functions which would have
far-reaching consequences if true.
05/2013;
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ABSTRACT: We study sorting machines consisting of a stack and a pop stack in series,
with or without a queue between them. While there are, a priori, four such
machines, only two are essentially different: a pop stack followed directly by
a stack, and a pop stack followed by a queue and then by a stack. In the former
case, we obtain complete answers for the basis and enumeration of the sortable
permutations. In the latter case, we present several conjectures.
03/2013;
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ABSTRACT: The simple permutations in two permutation classes --- the 321-avoiding
permutations and the skew-merged permutations --- are enumerated using a
uniform method. In both cases, these enumerations were known implicitly, by
working backwards from the enumeration of the class, but the simple
permutations had not been enumerated explicitly. In particular, the enumeration
of the simple skew-merged permutations leads to the first truly structural
enumeration of this class as a whole. The extension of this method to a wider
collection of classes namely grid classes of infinite paths is discussed.
01/2013;
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ABSTRACT: Infinite antichains of permutations have long been used to construct
interesting permutation classes and counterexamples. We prove the existence and
detail the construction of infinite antichains with arbitrarily large growth
rates. As a consequence, we show that every proper permutation class is
contained in a class with a rational generating function. While this result
implies the conclusion of the Marcus-Tardos theorem, that theorem is used in
our proof.
12/2012;
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ABSTRACT: We enumerate three specific permutation classes defined by two forbidden
patterns of length four. The techniques involve inflations of geometric grid
classes.
09/2012;
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ABSTRACT: 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
$\kappa\approx2.20557$ (a specific algebraic integer at which infinite
antichains begin to appear). Using language- and order-theoretic methods, we
prove that the substitution closures of geometric grid classes are partially
well-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 partially well-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 $\kappa$ has a
rational generating function. This bound is tight as there are permutation
classes with growth rate $\kappa$ which have nonrational generating functions.
02/2012;
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ABSTRACT: A geometric grid class consists of those permutations that can be drawn on a
specified set of line segments of slope \pm1 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.
08/2011;
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ABSTRACT: A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic
to the disjoint union of a 3-element chain and a 1-element chain. These posets
are of interest because of their connection with interval orders and their
appearance in the $(\3+\1)$-free Conjecture of Stanley and Stembridge. The
dimension 2 posets $P$ are exactly the ones which have an associated
permutation $\pi$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and
$i$ comes before $j$ in the one-line notation of $\pi$. So we say that a
permutation $\pi$ is {\it $(\3+\1)$-free} or {\it $(\3+\1)$-avoiding} if its
poset is $(\3+\1)$-free. This is equivalent to $\pi$ avoiding the permutations
2341 and 4123 in the language of pattern avoidance. We give a complete
structural characterization of such permutations. This permits us to find their
generating function.
02/2011;
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ABSTRACT: A selection of points drawn from a convex polygon, no two with the same vertical or horizontal coordinate, yields a permutation in a canonical fashion. We characterise and enumerate those permutations which arise in this manner and exhibit some interesting structural properties of the permutation class they form. We conclude with a permutation analogue of the celebrated Happy Ending Problem.
Discrete Mathematics. 01/2011;
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ABSTRACT: 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.
07/2010;
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Inf. Process. Lett. 01/2009; 109:626-629.
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Electr. J. Comb. 01/2009; 16.
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Theor. Comput. Sci. 01/2008; 391:150-163.
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ABSTRACT: A simple permutation is one that does not map a nontrivial interval onto an interval. It was recently proved by Albert and Atkinson that a permutation class with only finitely simple permutations has an algebraic generating function. We extend this result to enumerate permutations in such a class satisfying additional properties, e.g., the even permutations, the involutions, the permutations avoiding generalised permutations, and so on.
09/2006;
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ABSTRACT: We introduce and characterise grid classes, which are natural generalisations of other well-studied permutation classes. This characterisation allows us to give a new, short proof of the Fibonacci dichotomy: the number of permutations of length n in a permutation class is either at least as large as the nth Fibonacci number or is eventually polynomial.
03/2006;
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ABSTRACT: We determine the Möbius function of a poset of compositions of an integer. In fact, we give two proofs of this formula, one using an involution and one involving discrete Morse theory. This composition poset turns out to be intimately connected with subword order, whose Möbius function was determined by Björner. We show that, using a generalization of subword order, we can obtain both Björner’s results and our own as special cases.
Journal of Algebraic Combinatorics 01/2006; 24(2):117-136. · 0.75 Impact Factor
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ABSTRACT: We bound several quantities related to the packing density of the patterns 1(L+1)L...2. These bounds sharpen results of B\'ona, Sagan, and Vatter and give a new proof of the packing density of these patterns, originally computed by Stromquist in the case L=2 and by Price for larger L. We end with comments and conjectures.
06/2004;
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ABSTRACT: We prove that it is decidable whether a finitely based permutation class contains infinitely many simple permutations, and establish an unavoidable substructure result for simple permutations: every sufficiently long simple permutation contains an alternation or oscillation of length k.
Theoretical Computer Science.
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ABSTRACT: A simple permutation is one that never maps a nontrivial contiguous set of indices contiguously. Given a set of permutations that is closed under taking subpermutations and contains only finitely many simple permutations, we provide a framework for enumerating subsets that are restricted by properties belonging to a finite “query-complete set.” Such properties include being even, being an alternating permutation, and avoiding a given generalised (blocked or barred) pattern. We show that the generating functions for these subsets are always algebraic, thereby generalising recent results of Albert and Atkinson. We also apply these techniques to the enumeration of involutions and cyclic closures.
Journal of Combinatorial Theory, Series A.
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ABSTRACT: We investigate the notion of almost avoiding a permutation: πalmost avoidsβ if one can remove a single entry from π to obtain a β-avoiding permutation.
Discrete Mathematics.
