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403
Varieties of Increasing Trees
, 1992
"... An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We ..."
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Cited by 80 (7 self)
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An increasing tree is a labelled rooted tree in which labels along any branch from the root go in increasing order. Under various guises, such trees have surfaced as tree representations of permutations, as data structures in computer science, and as probabilistic models in diverse applications. We present a unified generating function approach to the enumeration of parameters on such trees. The counting generating functions for several basic parameters are shown to be related to a simple ordinary differential equation which is non linear and autonomous. Singularity analysis applied to the intervening generating functions then permits to analyze asymptotically a number of parameters of the trees, like: root degree, number of leaves, path length, and level of nodes. In this way it is found that various models share common features: path length is O(n log n), the distributions of node levels and number of leaves are asymptotically normal, etc.
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 78 (12 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.
equations for Hurwitz numbers
"... We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki. ..."
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Cited by 64 (3 self)
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We consider ramified coverings of P 1 with arbitrary ramification type over 0, ∞ ∈ P 1 and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a τfunction for the Toda lattice hierarchy of Ueno and Takasaki.
Enumeration of Planar Constellations
 Adv. in Appl. Math
, 2000
"... The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n ? 1, m ? 2, and let oe 0 be a permutation of Sn having d i cycles of length i, for i ? 1. We prove that the number of mtuples (oe 1 ; : : : ; oe m ) of permut ..."
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Cited by 62 (4 self)
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The enumeration of transitive ordered factorizations of a given permutation is a combinatorial problem related to singularity theory. Let n ? 1, m ? 2, and let oe 0 be a permutation of Sn having d i cycles of length i, for i ? 1. We prove that the number of mtuples (oe 1 ; : : : ; oe m ) of permutations of Sn such that: ffl oe 1 oe 2 \Delta \Delta \Delta oe m = oe 0 , ffl the group generated by oe 1 ; : : : ; oe m acts transitively on f1; 2; : : : ; ng, ffl P m i=0 c(oe i ) = n(m \Gamma 1) + 2, where c(oe i ) denotes the number of cycles of oe i , is m [(m \Gamma 1)n \Gamma 1]! [(m \Gamma 1)n \Gamma c(oe0 ) + 2]! Y i?1 " i / mi \Gamma 1 i !# d i : A onetoone correspondence relates these mtuples to some rooted planar maps, which we call constellations and enumerate via a bijection with some bicolored trees. For m = 2, we recover a formula of Tutte for the number of Eulerian maps. The proof extends the method applied in [21] to the latter case, and relies on the idea...
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
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Cited by 61 (14 self)
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I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
Asymptotics of Numbers of Branched Coverings of a Torus and Volumes of Moduli Spaces of Holomorphic Differentials
, 2000
"... ..."
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that i ..."
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Cited by 57 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.