scholarly journals Pattern Avoidance Classes and Subpermutations

10.37236/1957 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
M. D. Atkinson ◽  
M. M. Murphy ◽  
N. Ruškuc
Keyword(s):  
Generating Function ◽  
Structure Theorem ◽  
Pattern Avoidance ◽  
Natural Numbers ◽  
Direct Sums ◽  
Rational Generating Function

Pattern avoidance classes of permutations that cannot be expressed as unions of proper subclasses can be described as the set of subpermutations of a single bijection. In the case that this bijection is a permutation of the natural numbers a structure theorem is given. The structure theorem shows that the class is almost closed under direct sums or has a rational generating function.

A Rational Generating Function for Relative Divisors: 10750

2001 ◽  
Vol 108 (7) ◽  
pp. 670
Author(s):  
Leonard Smiley ◽  
David Callan ◽  
David M. Wells ◽  
Said Amghibech
Keyword(s):  
Generating Function ◽  
Rational Generating Function

scholarly journals Towards the Distribution of the Size of a Largest Planar Matching and Largest Planar Subgraph in Random Bipartite Graphs

10.37236/859 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Marcos Kiwi ◽  
Martin Loebl
Keyword(s):  
Generating Function ◽  
Maximum Size ◽  
Pattern Avoidance ◽  
Combinatorial Identities ◽  
Young Tableaux ◽  
Color Class ◽  
Lattice Walks ◽  
Type Property ◽  
Bivariate Generating Function ◽  
Standard Young Tableaux

We address the following question: When a randomly chosen regular bipartite multi–graph is drawn in the plane in the "standard way", what is the distribution of its maximum size planar matching (set of non–crossing disjoint edges) and maximum size planar subgraph (set of non–crossing edges which may share endpoints)? The problem is a generalization of the Longest Increasing Sequence (LIS) problem (also called Ulam's problem). We present combinatorial identities which relate the number of $r$-regular bipartite multi–graphs with maximum planar matching (maximum planar subgraph) of at most $d$ edges to a signed sum of restricted lattice walks in ${\Bbb Z}^d$, and to the number of pairs of standard Young tableaux of the same shape and with a "descend–type" property. Our results are derived via generalizations of two combinatorial proofs through which Gessel's identity can be obtained (an identity that is crucial in the derivation of a bivariate generating function associated to the distribution of the length of LISs, and key to the analytic attack on Ulam's problem). Finally, we generalize Gessel's identity. This enables us to count, for small values of $d$ and $r$, the number of $r$-regular bipartite multi-graphs on $n$ nodes per color class with maximum planar matchings of size $d$.Our work can also be viewed as a first step in the study of pattern avoidance in ordered bipartite multi-graphs.

scholarly journals Pattern avoidance in inversion sequences

2015 ◽  
Vol 25 (2) ◽  
pp. 157-176 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Generating Functions ◽  
Kernel Method ◽  
Fibonacci Numbers ◽  
Pattern Avoidance ◽  
Explicit Formulas ◽  
Schröder Numbers ◽  
Single Pattern ◽  
Combinatorial Methods

Abstract A permutation of length n may be represented, equivalently, by a sequence a1a2 • • • an satisfying 0 < ai < i for all z, which is called an inversion sequence. In analogy to the usual case for permutations, the pattern avoidance question is addressed for inversion sequences. In particular, explicit formulas and/or generating functions are derived which count the inversion sequences of a given length that avoid a single pattern of length three. Among the sequences encountered are the Fibonacci numbers, the Schröder numbers, and entry A200753 in OEIS. We make use of both algebraic and combinatorial methods to establish our results. An explicit Injection is given between two of the avoidance classes, and in three cases, the kernel method is used to solve a functional equation satisfied by the generating function enumerating the class in question.

scholarly journals Combinatorial specification of permutation classes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Frédérique Bassino ◽  
Mathilde Bouvel ◽  
Adeline Pierrot ◽  
Carine Pivoteau ◽  
Dominique Rossin
Keyword(s):  
Generating Function ◽  
Pattern Avoidance ◽  
System Of Equations ◽  
International Audience ◽  
Pattern Containment

International audience This article presents a methodology that automatically derives a combinatorial specification for the permutation class $\mathcal{C} = Av(B)$, given its basis $B$ of excluded patterns and the set of simple permutations in $\mathcal{C}$, 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 $\mathcal{C}$, this system being always positive and algebraic. It also yields a uniform random sampler of permutations in $\mathcal{C}$. The method presented is fully algorithmic. Cet article présente une méthodologie qui calcule automatiquement une spécification combinatoire pour la classe de permutations $\mathcal{C} = Av(B)$, étant donnés une base $B$ de motifs interdits et l’ensemble des permutations simples de $\mathcal{C}$, 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 $\mathcal{C}$, système qui est toujours positif et algébrique. Elle fournit aussi un générateur aléatoire uniforme de permutations dans $\mathcal{C}$. La méthode présentée est complètement algorithmique.

scholarly journals Random Orders and Gambler's Ruin

10.37236/1920 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Andreas Blass ◽  
Gábor Braun
Keyword(s):  
Random Process ◽  
Generating Function ◽  
Initial Segment ◽  
Linear Ordering ◽  
Natural Numbers ◽  
Gambler’S Ruin ◽  
Biased Coin

We prove a conjecture of Droste and Kuske about the probability that $1$ is minimal in a certain random linear ordering of the set of natural numbers. We also prove generalizations, in two directions, of this conjecture: when we use a biased coin in the random process and when we begin the random process with a specified ordering of a finite initial segment of the natural numbers. Our proofs use a connection between the conjecture and a question about the game of gambler's ruin. We exhibit several different approaches (combinatorial, probabilistic, generating function) to the problem, of course ultimately producing equivalent results.

scholarly journals Using Homological Duality in Consecutive Pattern Avoidance

10.37236/2005 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Anton Khoroshkin ◽  
Boris Shapiro
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Large Class ◽  
Generating Function ◽  
Generating Functions ◽  
Pattern Avoidance ◽  
Linear Ordinary Differential Equation ◽  
Sufficient Condition ◽  
Direct Algorithm ◽  
Polynomial Coefficients

Using an approach suggested by Dotsenko and Khoroshkin we present a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows us to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.

scholarly journals Enumeration of small Wilf classes avoiding 1342 and two other 4-letter patterns

2018 ◽  
Vol 27 (1) ◽  
pp. 62-97
Author(s):  
David Callan ◽  
Toufik Mansour
Keyword(s):  
Functional Equation ◽  
Generating Function ◽  
Pattern Avoidance ◽  
Standard Methods ◽  
Detailed Consideration ◽  
Symmetry Classes ◽  
Classical Pattern

Abstract This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

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