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39
Basic Analytic Combinatorics of Directed Lattice Paths
 Theoretical Computer Science
, 2001
"... This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then ess ..."
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Cited by 58 (10 self)
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This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.
Random planar graphs
 JOURNAL OF COMBINATORIAL THEORY, SERIES B 93 (2005) 187 –205
, 2005
"... We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive i ..."
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Cited by 45 (9 self)
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We study various properties of the random planar graph Rn, drawn uniformly at random from the class Pn of all simple planar graphs on n labelled vertices. In particular, we show that the probability that Rn is connected is bounded away from 0 and from 1. We also show for example that each positive integer k, with high probability Rn has linearly many vertices of a given degree, in each embedding Rn has linearly many faces of a given size, and Rn has exponentially many automorphisms.
Twenty combinatorial examples of asymptotics derived from multivariate generating functions
"... Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asym ..."
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Cited by 32 (14 self)
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Abstract. Let {ar: r ∈ Nd} be a ddimensional array of numbers for which the generating function F (z): = ∑ r arzr is meromorphic in a neighborhood of the origin. For example, F may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of F. Our purpose is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The survey begins by summarizing previous work on the asymptotics of univariate and multivariate generating functions. Next we describe the Morsetheoretic underpinnings of some new asymptotic techniques. We then quote and summarize these results in such a way that only elementary analyses are needed to check hypotheses and carry out computations. The remainder of the survey focuses on combinatorial applications, such as enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, and descents in permutations. After the individual examples, we discuss three broad classes of examples, namely, functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. These methods have
Growth and percolation on the uniform infinite planar triangulation
 Geom. Funct. Anal
, 2003
"... A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in [4], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a. ..."
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Cited by 26 (2 self)
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A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in [4], is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r 4 up to polylogarithmic factors, confirming heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r 2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an mgon (also defined in [4]) converges in distribution to an asymmetric stable random variable of type 1/2. By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability pc = 1/2 and that at pc percolation does not occur. Subject classification: Primary 05C80; Secondary 05C30, 82B43, 81T40. 1
Generating Labeled Planar Graphs Uniformly at Random
, 2003
"... We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3con ..."
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Cited by 25 (7 self)
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We present an expected polynomial time algorithm to generate a labeled planar graph uniformly at random. To generate the planar graphs, we derive recurrence formulas that count all such graphs with n vertices and m edges, based on a decomposition into 1, 2, and 3connected components. For 3connected graphs we apply a recent random generation algorithm by Schaeffer and a counting formula by Mullin and Schellenberg.
Exchangeable Gibbs partitions and Stirling triangles
"... For two collections of nonnegative and suitably normalised weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1,...,n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k W A1  · · ·W Ak, where Aj  is the number of ele ..."
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Cited by 22 (5 self)
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For two collections of nonnegative and suitably normalised weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1,...,n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k W A1  · · ·W Ak, where Aj  is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Πn of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [−∞, 1]. The case α = 1 is trivial, and for each value of α ̸ = 1 the set of possible Vweights is an infinitedimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the EwensPitman family of random partitions indexed by (α, θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α, θ)partition on the asymptotics of the number of blocks of Πn as n tends to infinity.
Analytic Urns
 March
, 2003
"... This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matri ..."
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Cited by 17 (1 self)
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This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matrix determines the replacement policy when a ball is drawn and its colour is observed.) The treatment starts from a quasilinear firstorder partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limits, and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x = 1. Several urn models, including a classical one associated with balanced trees (23 trees and fringebalanced search trees) and related to a previous study of Panholzer and Prodinger as well as all urns of balance 1 or 2, are shown to admit of explicit representations in terms of Weierstraß elliptic functions. Other consequences include a unification of earlier studies of these models and the detection of stable laws in certain classes of urns with an offdiagonal entry equal to zero.
Graph classes with given 3connected components: asymptotic enumeration and random graphs
, 2009
"... Consider a family T of 3connected graphs of moderate growth, and let G be the class of graphs whose 3connected components are graphs in T. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied c ..."
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Cited by 16 (7 self)
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Consider a family T of 3connected graphs of moderate growth, and let G be the class of graphs whose 3connected components are graphs in T. We present a general framework for analyzing such graphs classes based on singularity analysis of generating functions, which generalizes previously studied cases such as planar graphs and seriesparallel graphs. We provide a general result for the asymptotic number of graphs in G, based on the singularities of the exponential generating function associated to T. We derive limit laws, which are either normal or Poisson, for several basic parameters, including the number of edges, number of blocks and number of components. For the size of the largest block we find a fundamental dichotomy: classes similar to planar graphs have almost surely a unique block of linear size, while classes similar to seriesparallel graphs have only sublinear blocks. This dichotomy also applies to the size of the largest 3connected component. For some classes under study both regimes occur, because of a critical phenomenon as the edge density in the class varies.
Convolutions of inverse linear functions via multivariate residues
, 2004
"... lj(z1,...,zd) n j be the quotient of an analytic function by a product of linear functions lj: = 1 − � bijzi. We compute asymptotic formulae for the Taylor coefficients of F via the multivariate residue approach begun by [BM93]. By means of stratified Morse theory, we are able to give a short and f ..."
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Cited by 13 (7 self)
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lj(z1,...,zd) n j be the quotient of an analytic function by a product of linear functions lj: = 1 − � bijzi. We compute asymptotic formulae for the Taylor coefficients of F via the multivariate residue approach begun by [BM93]. By means of stratified Morse theory, we are able to give a short and fully implementable algorithm for determining an asymptotic series expansion.