# Edward A. Bender

University of California, San Diego, San Diego, California, United States

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## Publications (91)67.54 Total impact

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##### Article: Some Unsolved Problems in Map Enumeration
Edward A. Bender
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ABSTRACT: . This is a brief introduction to several problems related to the enumeration of maps in surfaces. The problems range from requests for bijective proofs to asymptotic problems to an algebraic question. 1. A Very Brief Introduction to Map Enumeration For many years, people have considered various problems in the enumeration of graphs, beginning with the enumeration of trees. Extensive work on the enumeration of maps did not begin until Tutte's work in the 1960's; however, one map enumeration problem--- the determination of the number of combinatorially inequivalent 3-dimensional convex polytopes---is over a century old. Traditionally an map has been thought of as a connected unlabeled graph which has been embedded in (= continuous injection into) a sphere. In the 1960's various classes of (rooted) maps were enumerated thanks to the pioneering work of Tutte. His approach consists of three main steps: 1. Enumerate rooted maps instead of those originally requested. A map is rooted if...
03/2000; 3.
• ##### Article: Submitted: August 3, 1999; Accepted: September 26, 1999
Edward A. Bender · E. Rodney Canfield
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ABSTRACT: Suppose that t 2 is an integer, and randomly label t graphs with the integers 1 : : : n. We give sufficient conditions for the number of edges common to all t of the labelings to be asymptotically Poisson as n ! 1. We show by example that our theorem is, in a sense, best possible. For Gn a sequence of graphs of bounded degree, each having at most n vertices, Tomescu [7] has shown that the number of spanning trees of Kn having k edges in common with Gn is asymptotically e Gamma2s=n (2s=n) k =k! Theta n nGamma2 , where s = s(n) is the number of edges in Gn . As an application of our Poisson-intersection theorem, we extend this result to the case in which maximum degree is only restricted to be O(n log log n= log n). We give an inversion theorem for falling moments, which we use to prove our Poisson-intersection theorem. AMS-MOS Subject Classification (1990): 05C30; Secondary: 05A16, 05C05, 60C05 the electronic journal of combinatorics 6 (1999), #R36 2 1. Introduction and St...
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##### Article: Intersections of Randomly Embedded Sparse Graphs are Poisson
Edward A. Bender · E. Rodney Canfield
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ABSTRACT: Suppose that t # 2 is an integer, and randomly label t graphs with the integers 1 ...n. We give sufficient conditions for the number of edges common to all t of the labelings to be asymptotically Poisson as n ##. We show by example that our theorem is, in a sense, best possible. For G n a sequence of graphs of bounded degree, each having at most n vertices, Tomescu [7] has shown that the number of spanning trees of K n having k edges in common with G n is asymptotically e -2s/n (2s/n) k /k! n n-2 , where s=s(n) is the number of edges in G n .As an application of our Poisson-intersection theorem, we extend this result to the case in which maximum degree is only restricted to be O(n log log n/ log n). We give an inversion theorem for falling moments, which we use to prove our Poisson-intersection theorem.
The electronic journal of combinatorics 10/1999; · 0.49 Impact Factor
• ##### Article: An Approximate Probabilistic Model for Structured Gaussian Elimination
Edward A. Bender · E. Rodney Canfield
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ABSTRACT: Introduction Modern algorithms [5, 7] for factoring integers or for solving discrete logarithms problems work in two phases. In the first one, one collects a huge amount of data that help create a large matrix that is triangularized in the second phase. For instance, the largest non-trivial number ever factored as of today is (10 211 Gamma 1)=9, which involved finding dependencies in a 4; 820; 249 Theta 4; 895; 741 boolean matrix (see [2]). The easiest way to solve the problem is to find a computer with enough memory so that the matrix fits in core and Gaussian elimination can be used. If such a behemoth is not available, alternative methods have to be used. A method that is widely used relies on the fact that the matrix we are interested in is sparse. For instance the matrix referred to above has only 48:1 non-zero coefficients per row on
Journal of Algorithms 05/1999; 31(2):271-290. DOI:10.1006/jagm.1999.1008 · 0.50 Impact Factor
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##### Article: The Number of Degree Restricted Rooted Maps on the Sphere
Edward A. Bender · E. Rodney Canfield
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ABSTRACT: Let D be a set of positive integers. Let m(n) be the number of n edged rooted maps on the sphere all of whose vertex degrees (or, dually, face degrees) lie in D. Using Brown's technique, we obtain the generating function for m(n) implicitly. We use it to prove that, when gcd(D) is even, m(n) C(D)n Gamma5=2 fl(D) n : It also yields known formulas for various special D. 2 Counting degree restricted maps Section 1: Introduction Let D be a set of positive integers containing some element exceeding 2, let M(x; y) = P i M i (x)y i be the generating function by edges and root face degree for rooted maps on the sphere such that each nonroot face degree lies in D and let m(n) be the number of n edged rooted maps all of whose face degrees lie in D. Define the coefficient operator with respect to y by [y k ] i P i0 f i (x)y i j = f k (x) and define [x k ] similarly. We will prove Theorem 1. There exist unique power series R 1 (x) and R 2 (x) such that R 1 = x 2 X i2D ...
SIAM Journal on Discrete Mathematics 03/1999; 7(1). DOI:10.1137/S0895480190177650 · 0.65 Impact Factor
• ##### Article: Submitted: March 27, 1998 Revised: January 5, 1999 Accepted: January 12, 1999
Edward A. Bender · L. Bruce Richmond
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ABSTRACT: An asymptotic estimate is given for the coefficients of products of large powers of generating functions. This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion. 1991 AMS Class. No. Primary: 41A63 Secondary: 05A16, 05C05, 41A60 the electronic journal of combinatorics 6 (1999), #R8 2 1. Introduction If f(0) 6= 0 has a (possibly formal) power series expansion at 0, the equation w = xf(w) determines the power series w(x). Two forms of the Lagrange inversion formula are gn = [x n ] g(w) = [x n ] i g(x)f(x) n f(1 Gamma xf 0 (x)=f(x)g j (1) = (1=n)[x n ] Gamma xg 0 (x)f(x) n Delta ; (2) where [x n ] h(x) denotes the coefficient of the monomial x n in the power series h(x). We obtained asymptotics for gn from (2) for some types of formal power series [6]. When f has a nonzero radius of convergence, various authors have studied the asymp...
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##### Article: Multivariate Asymptotics for Products of Large Powers with Applications to Lagrange Inversion
Edward A. Bender · L. Bruce Richmond
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ABSTRACT: An asymptotic estimate is given for the coe#cients of products of large powers of generating functions. This theorem and another local limit theorem which is useful for conditioning are applied to various combinatorial enumeration problems that involve multivariate Lagrange inversion. 1991 AMS Class. No. Primary: 41A63 Secondary: 05A16, 05C05, 41A60 ### ########### ###### ## ############# # ####### ### # 1. Introduction If f(0) #= 0 has a (possibly formal) power series expansion at 0, the equation w = xf(w) determines the power series w(x). Two forms of the Lagrange inversion formula are g n =[x n ] g(w)=[x n ] # g(x)f(x) n {(1 - xf # (x)/f (x)} # (1) =(1/n)[x n ] # xg # (x)f(x) n # , (2) where [x n ] h(x) denotes the coe#cient of the monomial x n in the power series h(x). We obtained asymptotics for g n from (2) for some types of formal power series [6]. When f has a nonzero radius of convergence, various authors have studied the asymptotics of [x n ] g...
The electronic journal of combinatorics 02/1999; 6(1). · 0.49 Impact Factor
• ##### Article: Submitted: March 3, 1998 Accepted: June 30, 1998
Edward A. Bender · L. Bruce Richmond
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ABSTRACT: The determinant that is present in traditional formulations of multivariate Lagrange inversion causes difficulties when one attempts (d+1) dGamma1 terms in contrast to the d! terms of the determinantal form. Thus it is likely to prove useful only for asymptotic purposes. 1991 AMS Classification Number. Primary: 05A15 Secondary: 05C05, 40E99 the electronic journal of combinatorics 5 (1998), #Rxx 2 1. Introduction Many researchers have studied the Lagrange inversion formula, obtaining a variety of proofs and extensions. Gessel [4] has collected an extensive set of references. For more recent results see Haiman and Schmitt [6], Goulden and Kulkarni [5], and Section 3.1 of Bergeron, Labelle, and Leroux [3]. Let boldface letters denote vectors and let a vector to a vector power be the product of componentwise exponentiation as in x n = x n1 1 Delta Delta Delta x nd d . Let [x n ] h(x) denote the coefficient of x n in h(x). Let ka i;j k denote the determinant of the d ...
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##### Article: Submap Density and Asymmetry Results for Two Parameter Map Families
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ABSTRACT: this paper, all limits are understood to be taken through those values of n
Combinatorics Probability and Computing 02/1999; 6(1). DOI:10.1017/S0963548396002799 · 0.62 Impact Factor
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##### Article: 0-1 Laws for Maps
Edward A. Bender · Kevin J. Compton · L. Bruce Richmond
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ABSTRACT: A class of finite structures has a 0--1 law with respect to a logic if every property expressible in the logic has a probability approaching a limit of 0 or 1 as the structure size grows. To formulate 0--1 laws for maps (i.e., embeddings of graphs in a surface), it is necessary to represent maps as logical structures. Three such representations are given, the most general being the full cross representation based on Tutte's theory of combinatorial maps. The main result says that if a class of maps has two properties, richness and large representativity, then the corresponding class of full cross representations has a 0--1 law with respect to first-order logic. As a corollary the following classes of maps on a surface of fixed type have a first-order 0--1 law: all maps, smooth maps, 2-connected maps, 3-connected maps, triangular maps, 2-connected triangular maps, and 3-connected triangular maps. c fl ??? John Wiley & Sons, Inc. Keywords: 0--1 law, maps 1. INTRODUCTION. In probability...
Random Structures and Algorithms 02/1999; 14(3). DOI:10.1002/(SICI)1098-2418(199905)14:33.0.CO;2-K · 0.92 Impact Factor
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##### Article: A Multivariate Lagrange Inversion Formula for Asymptotic Calculations
Edward A. Bender · L. Bruce Richmond
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ABSTRACT: The determinant that is present in traditional formulations of multivariate Lagrange inversion causes di#culties when one attempts to obtain asymptotic information. We obtain an alternate formulation as a sum of terms, thereby avoiding this di#- culty. 1991 AMS Classification Number. Primary: 05A15 Secondary: 05C05, 40E99 the electronic journal of combinatorics 5 (1998), #R33 2 1. Introduction Many researchers have studied the Lagrange inversion formula, obtaining a variety of proofs and extensions. Gessel [4] has collected an extensive set of references. For more recent results see Haiman and Schmitt [6], Goulden and Kulkarni [5], and Section 3.1 of Bergeron, Labelle, and Leroux [3]. Let boldface letters denote vectors and let a vector to a vector power be the product of componentwise exponentiation as in x n = x n1 1 x n d d .Let[x n ]h(x) denote the coe#cient of x n in h(x). Let #a i,j # denote the determinant of the d d matrix with entries a i,j . A traditional formul...
The electronic journal of combinatorics 08/1998; 5(1). · 0.49 Impact Factor
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##### Article: Connectedness, Classes, and Cycle Index
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ABSTRACT: This paper begins with the observation that half of all graphs containing no induced path of length 3 are disconnected. We generalize this in several directions. First, we give necessary and sufficient conditions (in terms of generating functions) for the probability of connectedness in a suitable class of graphs to tend to a limit strictly between zero and one. Next we give a general framework in which this and related questions can be posed, involving operations on classes of finite structures. Finally, we discuss briefly an algebra associated with such a class of structures, and give a conjecture about its structure. 1 1 Introduction The class of graphs containing no induced path of length 3 has many remarkable properties, stemming from the following well-known observation. Recall that an induced subgraph of a graph consists of a subset S of the vertex set together with all edges contained in S. Proposition 1.1 Let G be a finite graph with more than one vertex, containin...
Combinatorics Probability and Computing 06/1998; 8(1-2). DOI:10.1017/S0963548398003423 · 0.62 Impact Factor
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##### Article: The Fraction of Subspaces of GF(q)^n with a Specified Number of Minimal Weight Vectors is Asymptotically Poisson
Edward A. Bender · E. Rodney Canfield
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ABSTRACT: The weight of a vector in the finite vector space GF(q) n is the number of nonzero components it contains. We show that for a certain range of parameters (n; j; k; w) the number of k-dimensional subspaces having j(q Gamma 1) vectors of minimum weight w has asymptotically a Poisson distribution with parameter = Gamma n w Delta (q Gamma 1) wGamma1 q kGamman . As the Poisson parameter grows, the distribution becomes normal. AMS-MOS Subject Classification (1990). Primary: 05A16 Secondary: 05A15, 11T99 the electronic journal of combinatorics 4 (1997), #R3 2 1. Introduction Almost all the familiar concepts of linear algebra, such as dimension and linear independence, are valid without regard to the characteristic of the underlying field. An example of a characteristic-dependent result is that a nonzero vector cannot be orthogonal to itself; researchers accustomed to real vector spaces must modify their "intuition" on this point when entering the realm of finite fields. L...
The electronic journal of combinatorics 01/1998; · 0.49 Impact Factor
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##### Article: The Asymptotic Number of Labeled Graphs withnVertices,qEdges, and No Isolated Vertices
Edward A. Bender · E. Rodney Canfield · Brendan D. McKay
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ABSTRACT: Letd(n,q) be the number of labeled graphs withnvertices, q\le N={n \choose 2} edges, and no isolated vertices. Letx=q/nandk=2q−n. We determine functionswk∼1,a(x), andϕ(x) such that d(n,\ q)\sim w_k{N \choose q}\ e^{n\varphi (x)+a(x)} uniformly for allnandq>n/2.
Journal of Combinatorial Theory Series A 10/1997; 80(1):124-150. DOI:10.1006/jcta.1997.2798 · 0.78 Impact Factor
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##### Article: Log-Concavity and Related Properties of the Cycle Index Polynomials
Edward A. Bender · E.Rodney Canfield
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ABSTRACT: LetAndenote thenth-cycle index polynomial, in the variablesXj, for the symmetric group onnletters. We show that if the variablesXjare assigned nonnegative real values which are log-concave, then the resulting quantitiesAnsatisfy the two inequalitiesAn−1An+1⩽A2n⩽((n+1)/n)An−1An+1. This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever those ofg(u) satisfy a condition slightly weaker than log-concavity. The latter includes many familiar combinatorial sequences, only some of which were previously known to be log-concave. To prove the first inequality we show that in fact the differenceA2n−An−1An+1can be written as a polynomial with positive coefficients in the expressionsXjandXjXk−Xj−1Xk+1,j⩽k. The second inequality is proven combinatorially, by working with the notion of amarkedpermutation, which we introduce in this paper. The latter is a permutation each of whose cycles is assigned a subset of available markers {Mi, j}. Each marker has aweight, wt(Mi, j)=xj, and we relate the second inequality to properties of theweight enumerator polynomials. Finally, using asymptotic analysis, we show that the same inequalities hold fornsufficiently large when theXjare fixed with only finite many nonzero values, with no additional assumption on theXj.
Journal of Combinatorial Theory Series A 04/1996; 74(1):57–70. DOI:10.1006/jcta.1996.0037 · 0.78 Impact Factor
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##### Article: Asymptotic properties of rooted 3-connected maps on surfaces
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ABSTRACT: In this paper we obtain asymptotics for the number of rooted 3-connected maps on an arbitrary surface and use them to prove that almost all rooted 3-connected maps on any fixed surface have large edge-width and large face-width. It then follows from the result of Roberston and Vitray [10] that almost all rooted 3-connected maps on any fixed surface are minimum genus embeddings and their underlying graphs are uniquely embeddable on the surface.
Journal of the Australian Mathematical Society 01/1996; 60(01):31 - 41. DOI:10.1017/S144678870003737X · 0.14 Impact Factor
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##### Article: Admissible Functions and Asymptotics for Labelled Structures by Number of Components.
Edward A. Bender · L. Bruce Richmond
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ABSTRACT: Let a(n;k) denote the number of combinatorial structures of size n with k compo- nents. One often has P n;ka(n;k)x n yk=n !=e xp ' yC(x) " ,w hereC(x )i s frequently the exponential generating function for connected structures. How does a(n;k) behave as a function of k when n is large and C(x) is entire or has large singular- ities on its circle of convergence? The Flajolet-Odlyzko singularity analysis does not directly apply in such cases. We extend some of Hayman's work on admissi- ble functions of a single variable to functions of several variables. As applications, we obtain asymptotics and local limit theorems for several set partition problems, decomposition of vector spaces, tagged permutations, and various complete graph covering problems.
The electronic journal of combinatorics 01/1996; 3(1). · 0.49 Impact Factor
• ##### Article: Largest 4-Connected Components of 3-Connected Planar Triangulations.
Edward A. Bender · L. Bruce Richmond · Nicholas C. Wormald
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ABSTRACT: Let Tn be a 3-connected n-vertex planar triangulation chosen uniformly at random. Then the number of vertices in the largest 4-connected component of Tn is asymptotic to n/2 with probability tending to 1 as n → ∞. It follows that almost all 3-connected triangulations with n vertices have a cycle of length at least n/2 + o(n).
Random Structures and Algorithms 12/1995; 7(4):273-286. DOI:10.1002/rsa.3240070402 · 0.92 Impact Factor
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##### Article: Almost All Rooted Maps Have Large Representativity
Edward A. Bender · Zhicheng Gao · L. Bruce Richmond
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ABSTRACT: Let M be a map on a surface S. The edge-width of M is the length of a shortest non-contractible cycle of M . The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M . (The edge-width and face-width of a planar map may be defined to be infinity.) A map is an LEW-embedding if its maximum face valency is less than its edge-width. For several families of rooted maps on a given surface, we prove that there are positive constants c 1 and c 2 , depending on the family and the surface, such that 1. almost all maps with n edges have face-width and edge-width greater than c 1 log n and 2. the fraction of such maps which are LEW-embeddings and the fraction which are not LEW-embeddings both exceed n Gammac 2 . 2 1. Introduction We begin with some definitions: ffl A map is a connected graph G embedded in a surface S (a closed 2-manifold) such that all components of S Gamma G are simply connected regions. These components ar...
Journal of Graph Theory 10/1994; 18(6). DOI:10.1002/jgt.3190180603 · 0.63 Impact Factor
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##### Article: The Asymptotic Number of Rooted Maps on a Surface. II. Enumeration by Vertices and Faces.
Edward A. Bender · E. Rodney Canfield · L. Bruce Richmond
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ABSTRACT: We extend some of the earlier results on the enumeration of rooted maps on a surface by number of edges to simultaneous enumeration by vertices and faces. In particular, (i) an asymptotic formula is obtained, (ii) the generating functions on orientable surfaces are shown to be rational functions of the parameterizations of Arquès and of Tutte, and (iii) the generating function for rooted maps on the projective plane is given.
Journal of Combinatorial Theory Series A 07/1993; 63(2):318-329. DOI:10.1016/0097-3165(93)90063-E · 0.78 Impact Factor

#### Publication Stats

 2k Citations 67.54 Total Impact Points

#### Institutions

• ###### University of California, San Diego
• Department of Mathematics
San Diego, California, United States
• ###### University of Waterloo
• Department of Combinatorics & Optimization