Results 1  10
of
24
Grid Classes and the Fibonacci Dichotomy for Restricted Permutations
 Electronic J. Combinat
"... We introduce and characterise grid classes, which are natural generalisations of other wellstudied 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 ..."
Abstract

Cited by 24 (6 self)
 Add to MetaCart
(Show Context)
We introduce and characterise grid classes, which are natural generalisations of other wellstudied 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. 1
Simple permutations: decidability and unavoidable substructures
, 2006
"... We prove that it is decidable if 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. ..."
Abstract

Cited by 24 (6 self)
 Add to MetaCart
We prove that it is decidable if 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.
SMALL PERMUTATION CLASSES
, 2007
"... We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountab ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
(Show Context)
We establish a phase transition for permutation classes (downsets of permutations under the permutation containment order): there is an algebraic number κ, approximately 2.20557, for which there are only countably many permutation classes of growth rate (StanleyWilf limit) less than κ but uncountably many permutation classes of growth rate κ, answering a question of Klazar. We go on to completely characterize the possible subκ 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 wellorder, and atomicity (also known as the joint embedding property).
Finitely labeled generating trees and restricted permutations
 Journal of Symbolic Computation
, 2006
"... Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tre ..."
Abstract

Cited by 15 (5 self)
 Add to MetaCart
(Show Context)
Abstract. Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely many labels. Sometimes, however, this generating tree needs only finitely many labels. We characterize the finite sets of patterns for which this phenomenon occurs. We also present an algorithm — in fact, a special case of an algorithm of Zeilberger — that is guaranteed to find such a generating tree if it exists. 1.
Overview of some general results in combinatorial enumeration
, 2008
"... This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part ..."
Abstract

Cited by 11 (0 self)
 Add to MetaCart
(Show Context)
This survey article is devoted to general results in combinatorial enumeration. The first part surveys results on growth of hereditary properties of combinatorial structures. These include permutations, ordered and unordered graphs and hypergraphs, relational structures, and others. The second part advertises five topics in general enumeration: 1. counting lattice points in lattice polytopes, 2. growth of contextfree languages, 3. holonomicity (i.e., Precursiveness) of numbers of labeled regular graphs, 4. frequent occurrence of the asymptotics cn −3/2 r n and 5. ultimate modular periodicity of numbers of MSOLdefinable structures. 1
FINDING REGULAR INSERTION ENCODINGS FOR PERMUTATION CLASSES
"... We describe a practical algorithm which computes the accepting automaton for the insertion encoding of a permutation class, whenever this insertion encoding is regular. This algorithm is implemented in the accompanying Maple package INSENC, which can automatically compute the rational generating fun ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
We describe a practical algorithm which computes the accepting automaton for the insertion encoding of a permutation class, whenever this insertion encoding is regular. This algorithm is implemented in the accompanying Maple package INSENC, which can automatically compute the rational generating functions for such classes. 1.
The enumeration of permutations avoiding 2143 and 4231,
 Pure Math. Appl. (PU.M.A.)
, 2011
"... Abstract We enumerate the pattern class Av(2143, 4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes. ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract We enumerate the pattern class Av(2143, 4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes.
Combinatorial specification of permutation classes †
"... Abstract. This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoid ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is achieved considering both pattern avoidance and pattern containment constraints in permutations. The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in C. The method presented is fully algorithmic. Résumé. Cet article présente une méthodologie qui calcule automatiquement une spécification combinatoire pour la classe de permutations C = Av(B), étant donnés une base B de motifs interdits et l’ensemble des permutations simples de C, lorsque ces deux ensembles sont finis. Ce résultat est obtenu en considérant à la fois des contraintes de motifs interdits et de motifs obligatoires dans les permutations. La spécification obtenue donne un système d’équations satisfait par la série génératrice de la classe C, système qui est toujours positif et algébrique. Elle fournit aussi un générateur aléatoire uniforme de permutations dans C. La méthode présentée est complètement algorithmique.
Algorithms for Permutation Statistics
, 2011
"... Two sequences u, v of n positive integers are order isomorphic if ui < uj if and only if vi < vj for all pairs (i, j) ∈ {1, 2,..., n} 2. A permutation π = π(1)π(2) · · · π(n) ∈ Sn is said to contain σ ∈ Sk as a pattern if there is some ktuple 1 ≤ i1 < i2 < · · · < ik ≤ n such ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
Two sequences u, v of n positive integers are order isomorphic if ui < uj if and only if vi < vj for all pairs (i, j) ∈ {1, 2,..., n} 2. A permutation π = π(1)π(2) · · · π(n) ∈ Sn is said to contain σ ∈ Sk as a pattern if there is some ktuple 1 ≤ i1 < i2 < · · · < ik ≤ n such that π(i1)π(i2) · · · π(ik) is order isomorphic to σ. The subsequence π(i1)π(i2) · · · π(ik) is called a copy of σ. This notion of pattern containment is generalized to include adjacency restrictions, i.e., conditions that demand ix + 1 = ix+1 for certain x ∈ {1, 2,..., k − 1}. A permutation statistic is a function f: ⋃ n Sn → C. The primary permutation statistics studied in this work are written in terms of the number of copies of a given pattern or patterns. The central concern of this thesis is to compute answers to problems of the following type: “Given patterns σ (1) ,..., σ (t) and nonnegative numbers k1, k2,..., kt, how many permutations in Sn have ki copies of σ (i) for each i? ” The techniques which apply will depend on the nature of the patterns σ (i) , as well as whether or not all ki = 0.