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Analytic Analysis of Algorithms
, 1992
"... . The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial struct ..."
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Cited by 314 (11 self)
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. The average case analysis of algorithms can avail itself of the development of synthetic methods in combinatorial enumerations and in asymptotic analysis. Symbolic methods in combinatorial analysis permit to express directly the counting generating functions of wide classes of combinatorial structures. Asymptotic methods based on complex analysis permit to extract directly coefficients of structurally complicated generating functions without a need for explicit coefficient expansions. Three major groups of problems relative to algebraic equations, differential equations, and iteration are presented. The range of applications includes formal languages, tree enumerations, comparisonbased searching and sorting, digital structures, hashing and occupancy problems. These analytic approaches allow an abstract discussion of asymptotic properties of combinatorial structures and schemas while opening the way for automatic analysis of whole classes of combinatorial algorithms. I...
GFUN: A Maple Package for the Manipulation of Generating and Holonomic Functions in One Variable
, 1992
"... We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package. ..."
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Cited by 156 (19 self)
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We describe the gfun package which contains functions for manipulating sequences, linear recurrences or di erential equations and generating functions of various types. This document isintended both as an elementary introduction to the subject and as a reference manual for the package.
Noncommutative Elimination in Ore Algebras Proves Multivariate Identities
 J. SYMBOLIC COMPUT
, 1996
"... ... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation. ..."
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Cited by 98 (11 self)
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... In this article, we develop a theory of @finite sequences and functions which provides a unified framework to express algorithms proving and discovering multivariate identities. This approach is vindicated by an implementation.
Exact Enumeration Of 1342Avoiding Permutations A Close Link With Labeled Trees And Planar Maps
, 1997
"... Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of inde ..."
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Cited by 92 (7 self)
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Solving the first nonmonotonic, longerthanthree instance of a classic enumeration problem, we obtain the generating function H(x) of all 1342avoiding permutations of length n as well as an exact formula for their number Sn (1342). While achieving this, we bijectively prove that the number of indecomposable 1342avoiding permutations of length n equals that of labeled plane trees of a certain type on n vertices recently enumerated by Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number of rooted bicubic maps enumerated by Tutte in 1963. Moreover, H(x) turns out to be algebraic, proving the first nonmonotonic, longerthanthree instance of a conjecture of Zeilberger and Noonan. We also prove that n p Sn (1342) converges to 8, so in particular, limn!1 (Sn (1342)=Sn (1234)) = 0.
Generating Functions for Generating Trees
 PROCEEDINGS OF 11TH FORMAL POWER SERIES AND ALGEBRAIC COMBINATORICS
, 1999
"... Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumerati ..."
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Cited by 73 (18 self)
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Certain families of combinatorial objects admit recursive descriptions in terms of generating trees: each node of the tree corresponds to an object, and the branch leading to the node encodes the choices made in the construction of the object. Generating trees lead to a fast computation of enumeration sequences (sometimes, to explicit formulae as well) and provide efficient random generation algorithms. We investigate the links between the structural properties of the rewriting rules defining such trees and the rationality, algebraicity, or transcendence of the corresponding generating function.
Hypergeometric solutions of linear recurrences with polynomial coefficients
 J. Symb. Comput
, 1992
"... We describe algorithm Hyper which can be used to find all hypergeometric solutions of linear recurrences with polynomial coefficients. Let K be a field of characteristic zero. We assume that K is computable, meaning that the elements of K can be finitely represented and that there exist algorithms f ..."
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Cited by 64 (11 self)
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We describe algorithm Hyper which can be used to find all hypergeometric solutions of linear recurrences with polynomial coefficients. Let K be a field of characteristic zero. We assume that K is computable, meaning that the elements of K can be finitely represented and that there exist algorithms for carrying out the field operations. Let KN denote the ring of all sequences over K, with addition and multiplication
Algorithmic Manipulations and Transformations of Univariate Holonomic Functions and Sequences
, 1996
"... Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be ..."
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Cited by 57 (0 self)
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Holonomic functions and sequences have the property that they can be represented by a finite amount of information. Moreover, these holonomic objects are closed under elementary operations like, for instance, addition or (termwise and Cauchy) multiplication. These (and other) operations can also be performed "algorithmically". As a consequence, we can prove any identity of holonomic functions or sequences automatically. Based on this theory, the author implemented a package that contains procedures for automatic manipulations and transformations of univariate holonomic functions and sequences within the computer algebra system Mathematica. This package is introduced in detail. In addition, we describe some different techniques for proving holonomic identities.
Four Classes of PatternAvoiding Permutations under one Roof: Generating Trees with Two Labels
, 2003
"... Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ..."
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Cited by 35 (5 self)
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Many families of patternavoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of 123avoiding permutations. The rewriting rule automatically gives a functional equation satis ed by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series.