68 research outputs found

    A Complete Grammar for Decomposing a Family of Graphs into 3-connected Components

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    Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family of graphs (with some stability conditions) in terms of the 3-connected subfamily. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar. As a main application we recover in a purely combinatorial way the analytic expression found by Gim\'enez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter.Comment: 39 page

    Restricted non-separable planar maps and some pattern avoiding permutations

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    Tutte founded the theory of enumeration of planar maps in a series of papers in the 1960s. Rooted non-separable planar maps are in bijection with West-2-stack-sortable permutations, beta(1,0)-trees introduced by Cori, Jacquard and Schaeffer in 1997, as well as a family of permutations defined by the avoidance of two four letter patterns. In this paper we give upper and lower bounds on the number of multiple-edge-free rooted non-separable planar maps. We also use the bijection between rooted non-separable planar maps and a certain class of permutations, found by Claesson, Kitaev and Steingrimsson in 2009, to show that the number of 2-faces (excluding the root-face) in a map equals the number of occurrences of a certain mesh pattern in the permutations. We further show that this number is also the number of nodes in the corresponding beta(1,0)-tree that are single children with maximum label. Finally, we give asymptotics for some of our enumerative results.Comment: 18 pages, 14 figure

    A Meinardus theorem with multiple singularities

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    Meinardus proved a general theorem about the asymptotics of the number of weighted partitions, when the Dirichlet generating function for weights has a single pole on the positive real axis. Continuing \cite{GSE}, we derive asymptotics for the numbers of three basic types of decomposable combinatorial structures (or, equivalently, ideal gas models in statistical mechanics) of size nn, when their Dirichlet generating functions have multiple simple poles on the positive real axis. Examples to which our theorem applies include ones related to vector partitions and quantum field theory. Our asymptotic formula for the number of weighted partitions disproves the belief accepted in the physics literature that the main term in the asymptotics is determined by the rightmost pole.Comment: 26 pages. This version incorporates the following two changes implied by referee's remarks: (i) We made changes in the proof of Proposition 1; (ii) We provided an explanation to the argument for the local limit theorem. The paper is tentatively accepted by "Communications in Mathematical Physics" journa

    Controlled non uniform random generation of decomposable structures

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    Consider a class of decomposable combinatorial structures, using different types of atoms \Atoms = \{\At_1,\ldots ,\At_{|{\Atoms}|}\}. We address the random generation of such structures with respect to a size nn and a targeted distribution in kk of its \emph{distinguished} atoms. We consider two variations on this problem. In the first alternative, the targeted distribution is given by kk real numbers \TargFreq_1, \ldots, \TargFreq_k such that 0 < \TargFreq_i < 1 for all ii and \TargFreq_1+\cdots+\TargFreq_k \leq 1. We aim to generate random structures among the whole set of structures of a given size nn, in such a way that the {\em expected} frequency of any distinguished atom \At_i equals \TargFreq_i. We address this problem by weighting the atoms with a kk-tuple \Weights of real-valued weights, inducing a weighted distribution over the set of structures of size nn. We first adapt the classical recursive random generation scheme into an algorithm taking \bigO{n^{1+o(1)}+mn\log{n}} arithmetic operations to draw mm structures from the \Weights-weighted distribution. Secondly, we address the analytical computation of weights such that the targeted frequencies are achieved asymptotically, i. e. for large values of nn. We derive systems of functional equations whose resolution gives an explicit relationship between \Weights and \TargFreq_1, \ldots, \TargFreq_k. Lastly, we give an algorithm in \bigO{k n^4} for the inverse problem, {\it i.e.} computing the frequencies associated with a given kk-tuple \Weights of weights, and an optimized version in \bigO{k n^2} in the case of context-free languages. This allows for a heuristic resolution of the weights/frequencies relationship suitable for complex specifications. In the second alternative, the targeted distribution is given by a kk natural numbers n1,…,nkn_1, \ldots, n_k such that n1+⋯+nk+r=nn_1+\cdots+n_k+r=n where r≥0r \geq 0 is the number of undistinguished atoms. The structures must be generated uniformly among the set of structures of size nn that contain {\em exactly} nin_i atoms \At_i (1≤i≤k1 \leq i \leq k). We give a \bigO{r^2\prod_{i=1}^k n_i^2 +m n k \log n} algorithm for generating mm structures, which simplifies into a \bigO{r\prod_{i=1}^k n_i +m n} for regular specifications

    Further results on random cubic planar graphs

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    We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.Peer ReviewedPostprint (author's final draft

    Random cubic planar graphs revisited

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    The goal of our work is to analyze random cubic planar graphs according to the uniform distribution. More precisely, let G be the class of labelled cubic planar graphs and let gn be the number of graphs with n verticesPostprint (author's final draft

    A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics

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    A ``hybrid method'', dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux's method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions--this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.Comment: 31 page

    Generating Functions For Kernels of Digraphs (Enumeration & Asymptotics for Nim Games)

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    In this article, we study directed graphs (digraphs) with a coloring constraint due to Von Neumann and related to Nim-type games. This is equivalent to the notion of kernels of digraphs, which appears in numerous fields of research such as game theory, complexity theory, artificial intelligence (default logic, argumentation in multi-agent systems), 0-1 laws in monadic second order logic, combinatorics (perfect graphs)... Kernels of digraphs lead to numerous difficult questions (in the sense of NP-completeness, #P-completeness). However, we show here that it is possible to use a generating function approach to get new informations: we use technique of symbolic and analytic combinatorics (generating functions and their singularities) in order to get exact and asymptotic results, e.g. for the existence of a kernel in a circuit or in a unicircuit digraph. This is a first step toward a generatingfunctionology treatment of kernels, while using, e.g., an approach "a la Wright". Our method could be applied to more general "local coloring constraints" in decomposable combinatorial structures.Comment: Presented (as a poster) to the conference Formal Power Series and Algebraic Combinatorics (Vancouver, 2004), electronic proceeding
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