## Asymptotic enumeration methods (1995)

Venue: | Handbook of Combinatorics |

Citations: | 109 - 0 self |

### BibTeX

@INPROCEEDINGS{Odlyzko95asymptoticenumeration,

author = {A. M. Odlyzko},

title = {Asymptotic enumeration methods},

booktitle = {Handbook of Combinatorics},

year = {1995},

pages = {1063--1229},

publisher = {Elsevier}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1789 | Random Graphs
- Bollob'as
- 1981
(Show Context)
Citation Context ... are theorems that prove this under certain conditions. 6.5. Moments and distributions The second moment method is a frequently used technique in probabilistic arguments, as is shown in Chapter ? and =-=[55, 108, 348]-=-. It is based on Chebyshev's inequality, which says 49 that if X is a real-valued random variable with finite second moment E(X 2 ), then Prob (|X -E(X)| # #|E(X)|) # E(X 2 ) -E(X) 2 # 2 E(X) 2 . (6.8... |

1687 |
The probabilistic method
- Spencer
- 1992
(Show Context)
Citation Context ...martingales, branching processes, and Brownian motion asymptotics have been brought to bear on this topic. General introductions and references to these topics can be found in Chapter ? as well as in =-=[5, 11, 20, 21, 27, 92, 93, 108, 258, 260, 262, 270]-=-. 16.3. Statistical physics There is an extensive literature in mathematical physics concerned with asymptotic enumeration, especially in Ising models of statistical mechanics and percolation methods.... |

1476 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...ion, especially in Ising models of statistical mechanics and percolation methods. Many of the methods are related to combinatorial enumeration. For an introduction to them, see Chapter ? or the books =-=[30, 226]-=-. 16.4. Classical applied mathematics There are many techniques, such as the ray method and the WKB method, that have been developed for solving di#erential and integral equations in what we might cal... |

1455 |
An Introduction to Probability Theory
- Feller
- 1971
(Show Context)
Citation Context ...singularities (i.e., ones that do not grow rapidly as the argument approaches the circle of convergence) and give asymptotic relations for the sum of coe#cients. References for Tauberian theorems are =-=[117, 154, 190, 212, 325]-=-. Their main advantage is generality and ease of use, as is shown 63 by the applications made to 0-1 laws in [77, 78, 79]. They can often be applied when the information about generating functions is ... |

849 |
A Course of Modern Analysis
- Whittaker, Watson
- 1927
(Show Context)
Citation Context ... n , the number of rooted labeled trees on n nodes, is n n-1 . \Xi Proof of a form of the Lagrange-Burmann theorem is given in Chapter ?. Extensive discussion, proofs, and references are contained in =-=[81, 173, 205, 375]-=-. Some additional recent references are [159, 208]. There exist generalizations of the Lagrange-Burmann formula to several variables [173, 169, 208]. The Lagrange-Burmann formula, as stated above, is ... |

799 | Probability: Theory and examples - Durrett - 1991 |

658 |
The Theory of Partitions
- Andrews
- 1984
(Show Context)
Citation Context ...involve some loss of accuracy as a penalty for simplicity. Sometimes, the tradeo#s are clear. Let p(n) denote the number of partitions of an integer n. The Rademacher convergent series representation =-=[13, 23]-=- for p(n) is valid for any n # 1: p(n) = # -1 2 -1/2 # # m=1 Am (n)m 1/2 d dv (# -1 v sinh(Cm -1 # v )) # # # v=n , (1.3) where C = #(2/3) 1/2 , # v = (v - 1/24) 1/2 , (1.4) and the Am (n) satisfy A 1... |

630 | The on-line encyclopedia of integer sequences
- Sloane
- 2007
(Show Context)
Citation Context ... a systematic procedure for constructing such R(n, k). To conclude this section, we mention that a useful resource when investigating sequences arising in combinatorial settings is the book of Sloane =-=[345, 346]-=-, which lists several thousand sequences and gives references for them. Section 17 mentions some software systems that are useful in asymptotics. 11 4. Basic estimates: factorials and binomial coe#cie... |

447 |
Asymptotics and special functions
- Olver
- 1974
(Show Context)
Citation Context ...tated above can often be improved by using special properties of the function g(x). For example, when g(x) is analytic in x, there are contour integrals for Rm that sometimes give good estimates (cf. =-=[315]). The Poi-=-sson summation formula states that # # n=-# f(n + a) = # # m=-# exp(2#ima) # # -# f(y) exp(-2#imy)dy (5.75) for "nice" functions f(x). The functions for which (5.75) holds include all contin... |

402 | Introduction to Analytic Number Theory - Apostol - 1976 |

369 | The Theory of Branching Processes - Harris - 1963 |

359 |
The Art of Computer Programming, Vol. 3: Sorting and Searching
- Knuth
- 1973
(Show Context)
Citation Context ... spite of its importance and growth, asymptotic enumeration has seldom been presented in combinatorial literature at a level other than that of a research paper. There are several books that treat it =-=[43, 81, 175, 177, 235, 236, 237, 377]-=-, but usually only briefly. The only comprehensive survey that is available is the excellent and widely quoted paper of Bender [33]. Unfortunately it is somewhat dated. Furthermore, the last two decad... |

356 |
Lectures on Functional Equations and their Applications
- Aczél
- 1966
(Show Context)
Citation Context ...tween small and large singularities is important in asymptotics because di#erent methods are used in the two cases. A simple closed contour # in the complex plane is given by a continuous mapping # : =-=[0, 1]-=- # C with the properties that #(0) = #(1), and that #(s) #= #(t) whenever 0 # ss# 1 and either s #= 0 or t #= 1. Intuitively, # is a closed path in the complex plane that does not intersect itself. Fo... |

345 |
Combinatorial Enumeration
- Goulden, Jackson
- 1983
(Show Context)
Citation Context ...e in P \ R. The principle of inclusion-exclusion says that N= (R) = # R#Q#P (-1) |Q\R| N# (Q) . (5.53) (This is a basic version of the principle. For more general results, proofs, and references, see =-=[81, 173, 351]-=-.) 24 Example 5.6. Derangements of n letters. Let X be the set of permutations of n letters, and suppose that P i , 1 # i # n, is the property that the i-th letter is fixed by a permutation, and P = {... |

322 |
An Introduction to Combinatorial Analysis
- Riordan
- 1958
(Show Context)
Citation Context ... the x j independent variables. Faa di Bruno's formula makes it possible to compute successive derivatives of functions such as log A(z) in terms of the derivatives of A(z). For further examples, see =-=[81, 335, 336]-=-. Faa di Bruno's formula is derivable in a straightforward way from the multinomial theorem. Composition of generating functions occurs frequently in combinatorics and analysis of algorithms. When it ... |

319 |
Polylogarithms and Associated Functions
- Lewin
- 1981
(Show Context)
Citation Context ...tion g(z) defined by Eq. (8.62) has a nice expansion in the closed disk |z| # 1. Since g(z) = -z + # # m=2 (-1) m-1 m {Li m (z m ) - z m } , (11.38) where the Li m (w) are the polylogarithm functions =-=[251]-=-, one can use the theory of the Li m (w). A simpler way to proceed is to note, for example, that # # k=2 z 2k k 2 = # # k=2 z 2k k(k - 1) + r(z) , (11.39) where r(z) = - # # k=2 z 2k k 2 (k - 1) , (11... |

317 |
Singularity analysis of generating functions
- Flajolet, Odlyzko
- 1990
(Show Context)
Citation Context ... for larger values of m this approach becomes cumbersome, and other methods, such as those of Section 12, are necessary. \Xi 106 11.1. Transfer theorems This section presents some results, drawn from =-=[135]-=-, that allow one to translate an asymptotic expansion of a generating function around its dominant singularity into an asymptotic expansion for the coe#cients in a direct way. These results are useful... |

310 |
The Art of Computer Programming. Vol. 2. Seminumerical Algorithms
- Knuth
- 1997
(Show Context)
Citation Context ... spite of its importance and growth, asymptotic enumeration has seldom been presented in combinatorial literature at a level other than that of a research paper. There are several books that treat it =-=[43, 81, 175, 177, 235, 236, 237, 377]-=-, but usually only briefly. The only comprehensive survey that is available is the excellent and widely quoted paper of Bender [33]. Unfortunately it is somewhat dated. Furthermore, the last two decad... |

299 |
Combinatorial Identities
- Riordan
- 1968
(Show Context)
Citation Context ...ced to use asymptotic methods to estimate this sum. Recognizing when some combinatorial identity might apply is not easy. The literature on this subject is huge, and some of the references for it are =-=[172, 174, 186, 216, 336]-=-. Many of the books listed in the references are useful for this purpose. Generating functions (see Section 6) are one of the most common and powerful tools for proving identities. Here we only mentio... |

292 |
Graphical Enumeration
- Harary, Palmer
- 1973
(Show Context)
Citation Context ... functional equation f(z) = G(z, f(z)). Such equations arise frequently in graphical enumeration, and there is a standard procedure invented by Polya and developed by Otter that is almost algorithmic =-=[188, 189]-=- and routinely leads to them. Typically G(z, w) is analytic in z and w in a small neighborhood of (0, 0). Zeros of analytic functions in more than one dimension are not isolated, and by the implicit f... |

268 | An Introduction to the Analysis of Algorithms
- Sedgewick, Flajolet
- 1996
(Show Context)
Citation Context ...erive the asymptotics behavior from the specification of the recurrence. More generally, one can analyze asymptotics of a much greater variety of generating functions. Flajolet, Salvy, and Zimmermann =-=[124, 139]-=- have written a powerful program for just such computations. Their system uses Maple to carry out most of the basic analytic computations. It contains a remarkable amount of automated expertise in rec... |

248 |
Seminumerical Algorithms
- Knuth
- 1998
(Show Context)
Citation Context ... spite of its importance and growth, asymptotic enumeration has seldom been presented in combinatorial literature at a level other than that of a research paper. There are several books that treat it =-=[43, 81, 175, 177, 235, 236, 237, 377]-=-, but usually only briefly. The only comprehensive survey that is available is the excellent and widely quoted paper of Bender [33]. Unfortunately it is somewhat dated. Furthermore, the last two decad... |

246 |
The art of computer programming, vol.1: fundamental algorithms, 2 nd ed
- Knuth
- 1975
(Show Context)
Citation Context |

213 | Analytic Inequalities - Mitrinovic, Vasic - 1970 |

212 |
Evolution of random search trees
- Mahmoud
- 1992
(Show Context)
Citation Context ...uation. This was shown by Devroye [92] (see also [93]) by an application of the theory of branching processes. For a detailed exposition of this method and other applications to similar problems, see =-=[270]-=-. The basic generating function approach that we have used in most of this chapter leads to functional iterations which have not been solved so far. 149 15.3. Di#erential and integral equations Sectio... |

172 | An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities
- Wilf, Zeilberger
- 1992
(Show Context)
Citation Context ...- b n-2 /b n-1 . (3.4) Therefore Gosper's algorithm should be applied only when a n /a n-1 is rational. The other recent development is the Wilf-Zeilberger method for proving combinatorial identities =-=[379, 380]-=-. Given a conjectured identity, it provides an algorithmic procedure for verifying it. This method succeeds in a surprisingly wide range of cases. Typically, to prove an identity of the form # k U(n, ... |

168 |
Divergent Series
- Hardy
- 1973
(Show Context)
Citation Context ...singularities (i.e., ones that do not grow rapidly as the argument approaches the circle of convergence) and give asymptotic relations for the sum of coe#cients. References for Tauberian theorems are =-=[117, 154, 190, 212, 325]-=-. Their main advantage is generality and ease of use, as is shown 63 by the applications made to 0-1 laws in [77, 78, 79]. They can often be applied when the information about generating functions is ... |

155 | Probability Approximations via the Poisson Clumping Heuristic
- Aldous
- 1989
(Show Context)
Citation Context ...martingales, branching processes, and Brownian motion asymptotics have been brought to bear on this topic. General introductions and references to these topics can be found in Chapter ? as well as in =-=[5, 11, 20, 21, 27, 92, 93, 108, 258, 260, 262, 270]-=-. 16.3. Statistical physics There is an extensive literature in mathematical physics concerned with asymptotic enumeration, especially in Ising models of statistical mechanics and percolation methods.... |

154 |
Asymptotic approximations of integrals
- Wong
- 1989
(Show Context)
Citation Context ...tions on the length of this chapter, we do not present a full discussion of it. For a complete and insightful introduction to this technique, the reader is referred to [63]. Many other books, such as =-=[110, 115, 315, 385]-=- also have extensive presentations. What this section does is to outline the method, show when and how it can be applied and what kinds of estimates it produces. Examples of proper and improper applic... |

150 |
Asymptotic Expansions for Ordinary Differential Equations
- Wasow
- 1985
(Show Context)
Citation Context ...roblem may not be reflected in the resulting differential equation, and there may not be anything nice about it.) There is an extensive literature on analytic solutions of differential equations (cf. =-=[205, 206, 207, 272, 368, 372]-=-), but it is not easy to apply in general. Singularities of analytic functions that satisfy linear differential equations with analytic coefficients are usually of only a few basic forms, and so the m... |

144 | Combinatorial aspects of continued fractions - Flajolet - 1980 |

142 |
Decision procedure for indefinite hypergeometric summation
- Gosper
- 1978
(Show Context)
Citation Context ...proving identities. Here we only mention two recent developments that are of significance for both theoretical and practical reasons. One is Gosper's algorithm for indefinite hypergeometric summation =-=[171, 175]-=-. Given a sequence a 1 , a 2 , . . ., Gosper's algorithm determines whether the sequence of partial sums b n = n # k=1 a k , n = 1, 2, . . . (3.3) has the property that b n /b n-1 is a rational functi... |

139 | Mathematical Analysis - Apostol - 1974 |

134 |
Central and local limit theorems applied to asymptotic enumeration
- Bender
- 1973
(Show Context)
Citation Context ...ons with large singularities. However, there is also an extensive literature on small singularities. The main thread connecting most of these works is that of central and local limit theorems. Bender =-=[32]-=- initiated this development in the setting of two-variable problems. We present some of his results, since they are simpler than the later and more general ones that will be mentioned at the end of th... |

129 | On the altitude of nodes in random trees - Meir, Moon - 1978 |

127 |
L.A.Shepp, A variational problem for random Young tableaux
- Logan
- 1977
(Show Context)
Citation Context ...ty.) Example 10.5. Permutations without long increasing subsequences. Let u k (n) be the number of permutations of {1, 2, . . . , n} that have no increasing subsequence of length > k. Logan and Shepp =-=[257]-=- and Vershik and Kerov [370] established by calculus of variations and combinatorics that the average value of the longest increasing subsequence in a random permutation is asymptotic to 2n 1/2 . Frie... |

126 |
Differentiability finite power series
- Stanley
- 1980
(Show Context)
Citation Context ... this section we briefly summarize some of their main properties. Asymptotic properties of P -recursive sequences will be discussed in Section 9.2. The main references for the results quoted here are =-=[254, 350]-=-. A formal power series f(z) = # # k=0 a k z k (6.81) is called di#erentiably finite, or D-finite, if the derivatives f (n) (z) = d n f(z) dz n , n # 0, span a finite-dimensional vector space over the... |

122 |
de Bruijn, Asymptotic Methods in Analysis
- G
- 1981
(Show Context)
Citation Context ...o as Laplace's method for sums (in analogy to Laplace's method for estimating integrals, mentioned in Section 5.5, which proceeds in a similar spirit). There is extensive discussion of this method in =-=[63]-=-. Example 5.2. Sums of the partition function. We estimate U n = n # k=1 p(k) k , (5.16) where p(k) is the number of partitions of k. Since any partition of m- 1, say one with c j parts of size j, can... |

119 |
Asymptotic Expansion
- Erdelyi
- 1956
(Show Context)
Citation Context ...sary to estimate the behavior of f(x) as x ##, with the functions g(t), h(t) and the limits of integration # and # held fixed. There is a substantial theory of such integrals, and good references are =-=[54, 63, 100, 315]-=-. The basic technique is usually referred to as Laplace's method, and consists of approximating the integrand by simpler functions near its maxima. This approach is similar to the one that is discusse... |

117 |
Log-concave and unimodal sequences in algebra, combinat orics , and geometry
- Stanley
- 1989
(Show Context)
Citation Context ...mates bounds on individual coe#cients. This approach will be presented in Section 13, in the discussion of central and local limit theorems. The basic references for unimodality and log-concavity are =-=[222, 352]-=-. For recent results, see also [56] and the references given there. All the results listed below can be found in those sources and the references they list. In the rest of this subsection we will cons... |

107 |
An Introduction to Chaotic Dynamical systems, Addison-Wesley Pub
- Devaney
- 2000
(Show Context)
Citation Context ...nduction. However, this is an exceptional instance, and already recurrences of the type z n+1 = z 2 n + c for c a complex constant lead to deep questions about the Mandelbrot set and chaotic behavior =-=[91]-=-. Since linear recurrences are well understood, the best that one can hope for when confronted with a nonlinear recurrence is that it might be reducible to a linear one. This works in many situations.... |

106 | Saddlepoint Approximations in Statistics - Daniels - 1954 |

100 |
Asymptotic Methods in Enumeration
- Bender
- 1974
(Show Context)
Citation Context ...re are several books that treat it [43, 81, 175, 177, 235, 236, 237, 377], but usually only briefly. The only comprehensive survey that is available is the excellent and widely quoted paper of Bender =-=[33]-=-. Unfortunately it is somewhat dated. Furthermore, the last two decades have also witnessed a flowering of asymptotic analysis of algorithms, which was pioneered and popularized by Knuth. Combinatoria... |

99 |
String overlaps, pattern matching, and nontransitive games
- Guibas, Odlyzko
- 1981
(Show Context)
Citation Context ...FA (z) = z -k CA (z)GA (z) . (6.43) Solving the two equations (6.41) and (6.43), we find that FA (z) satisfies (6.38), while GA (z) = z k z k + (1 - 2z)CA (z) . (6.44) The proof above follows that in =-=[182]-=-, except that [182] uses generating functions in z -1 , so the formulas look di#erent. Applications of the formulas (6.38) and (6.44) will be found later in this chapter, as well as in [182, 130]. Oth... |

98 |
Asymptotic values for degrees associated with strips of Young diagrams
- Regev
- 1981
(Show Context)
Citation Context ...re few diagrams and their influence can be estimated explicitly using Stirling's formula, although Selberg-type integrals are involved and the analysis is complicated. This analysis was done by Regev =-=[329]-=-, who proved more general results. Here we sketch another approach to the asymptotics of u k (n) for k fixed. It is based on a result of Gessel [161]. If U k (z) = # # n=0 u k (n)z 2n (n!) 2 , (10.27)... |

98 |
Asymptotics of the Plancherel measure on symmetric group and the limiting form of the Young tableaux
- Kerov, Versik
- 1977
(Show Context)
Citation Context ...ons without long increasing subsequences. Let u k (n) be the number of permutations of {1, 2, . . . , n} that have no increasing subsequence of length > k. Logan and Shepp [257] and Vershik and Kerov =-=[370]-=- established by calculus of variations and combinatorics that the average value of the longest increasing subsequence in a random permutation is asymptotic to 2n 1/2 . Frieze [149] has proved recently... |

97 |
The average height of binary trees and other simple trees
- Flajolet, Odlyzko
- 1982
(Show Context)
Citation Context ...hen leads to a determination of the distribution. However, the resulting estimates do not say much about heights far away from the mean. A more careful analysis of the behavior of e h (z) can be used =-=[126]-=- to show that if x = h/(2n 1/2 ), then B h,n -B h-1,n B n # 2xn -1/2 # # m=1 m 2 (2m 2 x 2 - 3)e -m 2 x 2 (15.46) as n, h ##, uniformly for x = o((log n) 1/2 ), x -1 = o((log n) 1/2 ). For extremely s... |

96 | Average-case analysis of algorithms and data structures
- Vitter, Flajolet
- 1990
(Show Context)
Citation Context ...computer science, especially on average case analysis of algorithms, can therefore 5 be used fruitfully in asymptotic enumeration. One notable survey paper in that area is that of Vitter and Flajolet =-=[371]-=-. There are also presentations of relevant methods in the books [177, 209, 235, 236, 237, 223]. Section 18 is a guide to the literature on these topics. The aim of this chapter is to survey the most i... |

95 | Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Mathematica 68 - Pólya - 1937 |

85 |
Asymptotic enumeration of by degree sequence of graphs of high degree
- McKay, Wormald
- 1990
(Show Context)
Citation Context ...em is to apply this method when the dimension varies. A noteworthy case where this has been done successfully is the asymptotic enumeration of graphs with a given degree sequence by McKay and Wormald =-=[279]-=-. Example 13.3. Simple labeled graphs of high degree. Let G(n; d 1 , . . . , d n ) be the number of labeled simple graphs on n vertices with degree sequence d 1 , d 2 , . . . , d n . Then G(n; d 1 , .... |