Locally restricted compositions II: General restrictions, in preparation (2009)
Edward A. Bender, E. Rodney Canfield
We study compositions ⃗c = (c1,..., ck) of the integer n in which the value ci of the ith part is constrained based on previous parts within a fixed distance of ci. The constraints may depend on i...
Coefficients of Functional Compositions Often Grow Smoothly (2009)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond
Research supported by the NSA Mathematical Sciences Program. Research support by the NSERC. The coefficients of a power series A(x) are smooth if an−1/an approaches a limit. If A(x) = F (G(x)) and...
The map asymptotics constant tg (2009)
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond
Research supported by NSERC The constant tg appears in the asymptotic formulas for a variety of rooted maps on the orientable surface of genus g. Heretofore, studying this constant has been...
“And the winner is...” by (2008)
Let’s look at the simplest case first: two competitors, say the Comets and the Meteors. If they play each other enough, the better one will win more games and so we’ll know which is better. To...
Betting on Football Pools (2008)
There a two common types of pools. * One is among a group of friends and the person with the most correct guesses wins all the money that was bet. If several people have the same number of correct...
A multivariate Lagrange inversion formula for asymptotic calculations (2008)
Edward A. Bender, L. Bruce Richmond
The determinant that is present in traditional formulations of multivariate Lagrange inversion causes di culties when one attempts (d+1) d,1 terms in contrast to the d! terms of the determinantal...
A Discontinuity in the Distribution of Fixed Point Sums (2008)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
the electronic journal of combinatorics 10 (2003), #R15 1 The quantity f(n, r), defined as the number of permutations of the set [n] = {1,2,...n} whose fixed points sum to r, shows a sharp...
TEXed on 1/21/1999 Almost All Rooted Maps Have Large Representativity (2008)
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond
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...
Edward A. Bender, William J. Helton, L. Bruce Richmond
Ehrenborg obtained asymptotic results for nearly alternating permutations and conjectured an asymptotic formula for the number of permutations that have a nearly periodic run pattern. We prove a...
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k be a pair of compositions of n into k positive parts. We say this pair is irreducible if there is no positive j < k for which b1...
/res/papers/maps/FDEG.TEX 4/24/2003 The number of degree restricted rooted maps on the sphere (2008)
Edward A. Bender, E. Rodney Canfield
Running head: Counting degree restricted maps
Séminaire Lotharingien de Combinatoire 52 (2004), Article B50h Irreducible compositions (2008)
Edward A. Bender, Gregory F. Lawler, Robin Pemantle, Herbert S. Wilf
and the first return to the origin of a random walk Abstract. Let n = b1 + · · · + bk = b ′ 1 + · · · + b ′ k
The weight ofavector in the nite 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...
Edward A. Bender, E. Rodney Canfield
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...
The Fraction of Subspaces of (2007)
Gf With Specified, Edward A. Bender, E. Rodney Canfield
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...
The Asymptotic Number of Labeled Graphs With (2007)
Vertices Edges And, Edward A. Bender, Brendan D. Mckay, E. Rodney Canfield, E. Rodney Canfield
Let d(n; q) be the number of labeled graphs with n vertices, q N = \Gamma n 2 \Delta edges, and no isolated vertices. Let x = q=n and k = 2q \Gamma n. We determine functions w k 1, a(x), and...
Submitted: August 3, 1999; Accepted: September 26, 1999 (2007)
Te Mb Er, Edward A. Bender, E. Rodney Canfield
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...
Edward A. Bender, E. Rodney Canfield
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...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
The quantity f(n, r), defined as the number of permutations of the set [n] = 2, . . . n} whose fixed points sum to r, shows a sharp discontinuity in the neighborhood of r = n. We explain this...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
the electronic journal of combinatorics 10 (2003), #R15 1 The quantity f(n, r), defined as the number of permutations of the set [n] = {1, 2,...n} whose fixed points sum to r, shows a sharp...
A Discontinuity in the Distribution of Fixed Point Sums (2007)
Edward A. Bender, E. Rodney Canfield, L. Bruce Richmond, Herbert S. Wilf
The quantity f(n; r), de ned as the number of permutations of the set [n] = f1; 2; : : : ng whose xed points sum to r, shows a sharp discontinuity in the neighborhood of r = n. We explain this...
An Asymptotic Expansion for the Coefficients of Some Formal Power Series (2006)
Suppose A(x) is a formal power series with coefficients growing in an appropriately rapid fashion. If the coefficients of the formal power series F(x, y) do not grow too rapidly, then the...
Locally restricted compositions, I. Restricted adjacent differences, Electron (2005)
Edward A. Bender, E. Rodney Canfield
We study compositions of the integer n in which the first part, successive differences, and the last part are constrained to lie in prescribed sets L, D, R, respectively. A simple condition on D...
Irreducible compositions and the first return to the origin of a random walk (2004)
Bender, Edward A., Lawler, Gregory F., Pemantle, Robin, Wilf, Herbert S.
Let $n = b_1 + ... + b_k = b_1' + \cdot + b_k'$ be a pair of compositions of $n$ into $k$ positive parts. We say this pair is {\em irreducible} if there is no positive $j < k$ for which $b_1 + ......
A Discontinuity in the Distribution of Fixed Point Sums (2003)
Bender, Edward A., Canfield, E. Rodney, Richmond, L. Bruce, Wilf, Herbert S.
The quantity $f(n,r)$, defined as the number of permutations of the set $[n]=\{1,2,... n\}$ whose fixed points sum to $r$, shows a sharp discontinuity in the neighborhood of $r=n$. We explain this...
Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls (2003)
Edward A. Bender, William J. Helton, L. Bruce Richmond
Ehrenborg obtained asymptotic results for nearly alternating permutations and conjectured an asymptotic formula for the number of permutations that have a nearly periodic run pattern. We prove a...
Asymptotics of Permutations with Nearly Periodic Patterns of Rises and Falls (2003)
Edward A. Bender, William J. Helton, L. Bruce Richmond
Ehrenborg obtained asymptotic results for nearly alternating permutations and conjectured an asymptotic formula for the number of permutations that have a nearly periodic run pattern. We prove a...
Similarity to Symmetric Matrices over Fields Which are not Formally Real. (2002)
It is shown that a matrix is similar to a symmetric matrix over a field of characteristic 2 if and only if the minimum polynomial of the matrix is not the product of distinct irreducible polynomials...
The number of labeled 2-connected planar graphs (2000)
Edward A. Bender, Zhicheng Gao, Nicholas C. Wormald
We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many...
The number of labeled 2-connected planar graphs (2000)
Edward A. Bender, Zhicheng Gao, Nicholas C. Wormald
We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many...
The number of labeled 2-connected planar graphs (2000)
Edward A. Bender, Zhicheng Gao, Nicholas C. Wormald
We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many...
Asymptotics for the Probability of Connectedness and the Distribution of Number of Components (2000)
Jason P. Bell, Edward A. Bender, Peter J. Cameron, L. Bruce Richmond
Let ae n be the fraction of structures of "size" n which are "connected"; e.g., (a) the fraction of labeled or unlabeled n-vertex graphs having one component, (b) the fraction of...
The Number of Labeled 2-Connected Planar Graphs (2000)
Edward A. Bender, Zhicheng Gao, Nicholas C. Wormald
We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many...
Jason P. Bell, Edward A. Bender, Peter J. Cameron, L. Bruce Richmond
Let n be the fraction of structures of \size " n which are \connected"; e.g., (a) the fraction of labeled or unlabeled n-vertex graphs having one component, (b) the fraction of...
The Number of Labeled 2-Connected Planar Graphs (2000)
Edward A. Bender, Zhicheng Gao, Nicholas C. Wormald
We derive the asymptotic expression for the number of labeled 2-connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many...
Jason P. Bell, Edward A. Bender, Peter J. Cameron, L. Bruce Richmond
Let ρn be the fraction of structures of “size ” n which are “connected”; e.g., (a) the fraction of labeled or unlabeled n-vertex graphs having one component, (b) the fraction of partitions...
Multivariate asymptotics for products of large powers with applications to Lagrange inversion (1999)
Edward A. Bender, L. Bruce Richmond
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...
Log-Concavity and Related Properties of the Cycle Index Polynomials (1999)
Edward A. Bender, E. Rodney Canfield
Let An denote the n-th cycle index polynomial, in the variables X j , for the symmetric group on n letters. We show that if the variables X j are assigned nonnegative real values which are...
Edward A. Bender, Kevin J. Compton, L. Bruce Richmond
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...
Asymptotic Properties of Labeled Connected Graphs (1999)
Edward A. Bender, E. Rodney Canfield, Brendan D. Mckay
We prove various properties of C(n; q), the set of n-vertex q- edge labeled connected graphs. The domain of validity of the asymptotic formula of Erdos and R'enyi for jC(n; q)j is extended and...
The Number of Degree Restricted Rooted Maps on the Sphere (1999)
Edward A. Bender, E. Rodney Canfield
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...
Almost All Rooted Maps Have Large Representativity (1999)
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond
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...
Submitted: March 27, 1998 Revised: January 5, 1999 Accepted: January 12,1999 (1999)
An Ua Ry, Edward A. Bender, L. Bruce Richmond
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...
Intersections of Randomly Embedded Sparse Graphs are Poisson (1999)
Edward A. Bender, E. Rodney Canfield
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...
Intersections of Randomly Embedded Sparse Graphs are Poisson (1999)
Edward A. Bender, E. Rodney Canfield
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...
Multivariate Asymptotics for Products of Large Powers with Applications to Lagrange Inversion (1999)
Edward A. Bender, L. Bruce Richmond
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...
Edward A. Bender, Kevin J. Compton, L. Bruce Richmond
A class of nite 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...
Please address communications to: (1999)
Edward A. Bender, Brendan D. Mckay
The asymptotic number of labeled graphs with
Intersections of Randomly Embedded Sparse Graphs are Poisson (1999)
Suppose that t 2isaninteger, and randomly label t graphs with the integers 1:::n. We give su cient conditions for the number of edges common to all t of the labelings to be asymptotically Poisson as...
Multivariate asymptotics for products of large powers with applications to Lagrange inversion (1999)
Edward A. Bender, L. Bruce Richmond
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...
THE ENUMERATIVE USES OF GENERATING FUNCTIONS. (1998)
Bender,Edward A., Goldman,Jay R.
'Objects' may be viewed as a product of 'prime objects' in such a way that the appropriate type of generating function is immediately suggested by the combinatorial problem. Many graph theory...
Research in Enumeration and Graph Theory with Applications to Computer Science. (1998)
During the contract, almost half of the author's effort has been directed toward studying maps on surfaces other than the sphere. Other completed research was on convex polyhedra, and the size of...
A Multivariate Lagrange Inversion Formula for Asymptotic Calculations (1998)
Edward A. Bender, L. Bruce Richmond
The determinant that is present in traditional formulations of multivariate Lagrange inversion causes difficulties when one attempts (d+1) d\Gamma1 terms in contrast to the d! terms of the...
Ed Ccrwest Org, Edward A. Bender, S. Gill Williamson
Let # and # be a partition of {1, . . . , n} into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the input...
A Multivariate Lagrange Inversion Formula for Asymptotic Calculations (1998)
Edward A. Bender, L. Bruce Richmond
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...
A Multivariate Lagrange Inversion Formula for Asymptotic Calculations (1998)
Edward A. Bender, L. Bruce Richmond
The determinant that is present in traditional formulations of multivariate Lagrange inversion causes difficulties when one attempts to obtain asymptotic information. We obtain an alternate...
Periodic sorting using minimum delay, recursively constructed merging networks (1998)
Edward A. Bender, S. Gill Williamson
Let and be a partition of f1;:::;ng into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the input sequence...
Submap Density and Asymmetry Results for Two Parameter Map Families (1997)
Edward A. Bender, E. Rodney Canfield, Zhicheng Gao, L. Bruce Richmond
this paper, all limits are understood to be taken through those values of n
Periodic Sorting Using Minimum Delay, Recursively Constructed Merging Networks (1997)
Edward A. Bender, S. Gill Williamson
Let ff and fi be a partition of f1; : : : ; ng into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the...
Periodic Sorting Using Minimum Delay, Recursively Constructed Merging Networks (1997)
Edward A. Bender, S. Gill Williamson
Let # and # be a partition of {1,...,n} into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the input...
Submap density and asymmetry results for two parameter map families (1997)
Edward A. Bender, Zhicheng Gao, L. Bruce Richmond
2. Research supported by NSA and NSF.
Ed Ccrwest Org, Edward A. Bender, L. Bruce Richmond
Let a(n; k) denote the number of combinatorial structures of size n with k components. One often has P n;k a(n; k)x n y k =n! = exp \Phi yC(x) \Psi , where C(x) is frequently the exponential...
Admissible Functions and Asymptotics for Labelled Structures by Number of Components (1996)
Edward A. Bender, L. Bruce Richmond
Let a(n; k) denote the number of combinatorial structures of size n with k components. One often has P n;k a(n; k)x n y k =n! = exp \Phi yC(x) \Psi , where C(x) is frequently the exponential...
The Asymptotic Number of Labeled Graphs with n Vertices, q Edges, and No Isolated Vertices (1996)
Edward A. Bender, Brendan D. Mckay, E. Rodney Canfield, E. Rodney Canfield
Let d(n; q) be the number of labeled graphs with n vertices, q N = \Gamma n 2 \Delta edges, and no isolated vertices. Let x = q=n and k = 2q \Gamma n. We determine functions w k ¸ 1, a(x), and...
Edward A. Bender, E. Rodney Canfield
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...
Admissible functions and asymptotics for labelled structures by number of components (1996)
Edward A. Bender, L. Bruce Richmond
Let a(n; k) denote the number of combinatorial structures of size n with k components. One often has P n;k a(n; k)xn y k =n! = exp yC(x) , where C(x) is frequently the exponential generating function...
Log-concavity and related properties of the cycle index polynomials (1996)
Edward A. Bender, E. Rodney Canfield
Log-concavity and cycle index polynomials Let An denote the n-th cycle index polynomial, in the variables Xj, for the symmetric group on n letters. We show that if the variables Xj are assigned...
Log-concavity and related properties of the cycle index polynomials (1996)
Log-concavity and cycle index polynomials LetAn denote then-th cycle index polynomial, in the variablesXj, for the symmetric group onnletters. We showthatif the variablesXj are assigned nonnegative...
Asymptotic properties of labeled connected graphs (1992)
Edward A. Bender, E. Rodney Canfield, Brendan D. Mckay
notwithstanding any copyright notation hereon. Properties of Labeled Connected Graphs We prove various properties of C(n, q), the set of n-vertex qedge labeled connected graphs. The domain of...
Some Unsolved Problems in Map Enumeration (1991)
. 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...
An introduction to mathematical modeling / Edward A. Bender (1978)
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