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177
Boltzmann Samplers For The Random Generation Of Combinatorial Structures
 Combinatorics, Probability and Computing
, 2004
"... This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combina ..."
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Cited by 67 (2 self)
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This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class  an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on realarithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.
Boltzmann sampling of unlabelled structures
"... Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they ..."
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Cited by 22 (8 self)
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Boltzmann models from statistical physics combined with methods from analytic combinatorics give rise to efficient algorithms for the random generation of unlabelled objects. The resulting algorithms generate in an unbiased manner discrete configurations that may have nontrivial symmetries, and they do so by means of realarithmetic computations. We present a collection of construction rules for such samplers, which applies to a wide variety of combinatorial classes, including integer partitions, necklaces, unlabelled functional graphs, dictionaries, seriesparallel circuits, term trees and acyclic molecules obeying a variety of constraints, and so on. Under an abstract realarithmetic computation model, the algorithms are, for many classical structures, of linear complexity provided a small tolerance is allowed on the size of the object drawn. As opposed to many of their discrete competitors, the resulting programs routinely make it possible to generate random objects of sizes in the range 10⁴ –10⁶.
The repulsive lattice gas, the independentset polynomial, and the Lovász local lemma
, 2004
"... We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independent ..."
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Cited by 20 (6 self)
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We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {px} if and only if the independentset polynomial for G is nonvanishing in the polydisc of radii {px}. Furthermore, we show that the usual proof of the Lovász local lemma — which provides a sufficient condition for this to occur — corresponds to a simple inductive argument for the nonvanishing of the independentset polynomial in a polydisc, which was discovered implicitly by Shearer [98] and explicitly by Dobrushin [37, 38]. We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for “soft” dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternatingsign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.
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 & ..."
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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 &times; 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.
Categorified algebra and quantum mechanics
, 2006
"... Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a par ..."
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Cited by 13 (1 self)
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Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of [x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators ” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic
Boltzmann oracle for combinatorial systems
 In Algorithms, Trees, Combinatorics and Probabilities
, 2008
"... Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide s ..."
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Cited by 13 (5 self)
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Boltzmann random generation applies to welldefined systems of recursive combinatorial equations. It relies on oracles giving values of the enumeration generating series inside their disk of convergence. We show that the combinatorial systems translate into numerical iteration schemes that provide such oracles. In particular, we give a fast oracle based on Newton iteration.
Enumeration of MAry Cacti
"... The purpose of this paper is to enumerate various classes of cyclically colored mgonal plane cacti, called mary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explic ..."
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Cited by 13 (4 self)
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The purpose of this paper is to enumerate various classes of cyclically colored mgonal plane cacti, called mary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled mary cacti, according to i) the number of polygons, ii) the vertexcolor distribution, iii) the vertexdegree distribution of each color. We also enumerate mary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of mary cacti in terms of rooted and of pointed cacti. A variant of the mdimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to mary cacti. 1 Introduction A cactus is a connected simple graph in which each edge lies in exactly one elementary cycle. It ...
Random Sampling from Boltzmann Principles
, 2002
"... This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. ..."
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Cited by 12 (2 self)
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This extended abstract proposes a surprisingly simple framework for the random generation of combinatorial configurations based on Boltzmann models. Random generation of possibly complex structured objects is performed by placing an appropriate measure spread over the whole of a combinatorial class. The resulting algorithms can be implemented easily within a computer algebra system, be analysed mathematically with great precision, and, when suitably tuned, tend to be efficient in practice, as they often operate in linear time.
Effective scalar products of Dfinite symmetric series
 Journal of Combinatorial Theory Series A, 112:1
"... Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We ext ..."
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Cited by 12 (9 self)
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Abstract. Many combinatorial generating functions can be expressed as combinations of symmetric functions, or extracted as subseries and specializations from such combinations. Gessel has outlined a large class of symmetric functions for which the resulting generating functions are Dfinite. We extend Gessel’s work by providing algorithms that compute differential equations these generating functions satisfy in the case they are given as a scalar product of symmetric functions in Gessel’s class. Examples of applications to kregular graphs and Young tableaux with repeated entries are given. Asymptotic estimates are a natural application of our method, which we illustrate on the same model of Young tableaux. We also derive a seemingly new formula for the Kronecker product of the sum of Schur functions with itself. (This article completes the extended abstract published in the proceedings of FPSAC’02 under the title “Effective DFinite Symmetric Functions”.)
Feynman diagrams in algebraic combinatorics, Séminaire Lotharingien de Combinatoire 49
 2002–04), Article B49c, 45 pp. SPECIES AND FEYNMAN DIAGRAMS 37
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