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Enumerative Applications Of Symmetric Functions
 Proceedings of the 17th Séminaire Lotharingien, Publ. I.R.M.A. Strasbourg
, 1987
"... This paper consists of two related parts. In the first part the theory of Dfinite power series in several variables and the theory of symmetric functions are used to prove Precursiveness for regular graphs and digraphs and related objects, that is, that their counting sequences satisfy linear homo ..."
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This paper consists of two related parts. In the first part the theory of Dfinite power series in several variables and the theory of symmetric functions are used to prove Precursiveness for regular graphs and digraphs and related objects, that is, that their counting sequences satisfy linear homogeneous recurrences with polynomial coefficients. Previously this has been accomplished only for small degrees. See, for example, GOULDEN, JACKSON, and REILLY [7], GOULDEN and JACKSON [6], and READ [16, 18]. These authors found the recurrences satisfied by the sequences in question. Although the methods used here are in principle constructive, we are concerned here only with the question of existence of these recurrences and we do not find them. In the second part we consider a generalization of symmetric functions in several sets of variables, first studied by MACMAHON [13 ; 14, Vol. 2, pp. 280326]. MacMahon's generalized symmetric functions can be used to find explicit formulas and prove Precursiveness for some objects to which the theory of ordinary symmetric functions does not apply, such as Latin rectangles and 01 matrices with zeros on the diagonal and given row and column sums.
Balancing The nCube: A census of colorings
"... Weights of 1 or 0 are assigned to the vertices of the ncube in ndimensional Euclidean space. Such an ncube is called balanced if its center of mass coincides precisely with its geometric center. The seldomused nvariable form of P'olya's enumeration theorem is applied to express the n ..."
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Weights of 1 or 0 are assigned to the vertices of the ncube in ndimensional Euclidean space. Such an ncube is called balanced if its center of mass coincides precisely with its geometric center. The seldomused nvariable form of P'olya's enumeration theorem is applied to express the number N n;2k of balanced configurations with 2k vertices of vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers N n;2k depend on the numbers A n;2k of specially restricted configurations. A table of values of N n;2k and A n;2k is provided for n = 3; 4; 5 and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition. Key words. ncube, boolean functions, P'olya enumeration, supe...
Towards a stable definition of KolmogorovChaitin complexity
, 2008
"... Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the KolmogorovChaitin complexity of a string s. Some attempts have ..."
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Although information content is invariant up to an additive constant, the range of possible additive constants applicable to programming languages is so large that in practice it plays a major role in the actual evaluation of K(s), the KolmogorovChaitin complexity of a string s. Some attempts have been made to arrive at a framework stable enough for a concrete definition of K, independent of any constant under a programming language, by appealing to the naturalness of the language in question. The aim of this paper is to present an approach to overcome the problem by looking at a set of models of computation converging in output probability distribution such that that naturalness can be inferred, thereby providing a framework for a stable definition of K under the set of convergent models of computation.
Discrete Mathematics for Combinatorial Chemistry
, 1998
"... The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry. ..."
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The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.
Exponents of 2 in the Numbers of Unlabeled Graphs and Tournaments
, 1991
"... Let g(n) denote the number of unlabeled graphs on n nodes, and let e(n) denote its 2part, i.e., the exponent of the largest power of 2 which divides g(n). It is shown that for odd n 5, e(n) = (n + 1)=2 \Gamma blog 2 nc and for even n 4 e(n) n=2 \Gamma blog 2 nc with equality if, and only i ..."
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Let g(n) denote the number of unlabeled graphs on n nodes, and let e(n) denote its 2part, i.e., the exponent of the largest power of 2 which divides g(n). It is shown that for odd n 5, e(n) = (n + 1)=2 \Gamma blog 2 nc and for even n 4 e(n) n=2 \Gamma blog 2 nc with equality if, and only if, n is a power of 2. Similarly, let t(n) denote the number of unlabeled tournaments on n nodes and r(n) its 2part. It is shown that for all odd n, r(n) = (n \Gamma 1)=2 and for all even n 4 r(n) n=2 with equality if, and only if, '(n)=2 is odd. Here '(n) is the Euler totient function. A preliminary version of this paper (without tournament results) was presented at the 22nd Southeastern International Conference on Combinatorics, Graph Theory, and Computing in Baton Rouge, LA, on February 11, 1991. The present version appears in the proceedings of that conference, Congr. Numer. 82 (1991) 139155. 1 Introduction Let g(n) be the number of nonisomorphic graphs on n nodes, often ref...
Exact Enumeration and Sampling of Matrices with Specified Margins
"... We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or nonnegative integer matrices are handled. The method is distinguished by applic ..."
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We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or nonnegative integer matrices are handled. The method is distinguished by applicability to nonregular margins, tractability on large matrices, and the capacity for exact sampling.
Decomposition Characterizations of Classes of 2Connected Graphs
"... By applying the Tutte decomposition of 2connected graphs into 3block trees we provide unique structural characterizations of several classes of 2connected graphs, including minimally 2connected graphs, minimally 2edgeconnected graphs, critically 2connected graphs, critically 2edgeconnected ..."
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By applying the Tutte decomposition of 2connected graphs into 3block trees we provide unique structural characterizations of several classes of 2connected graphs, including minimally 2connected graphs, minimally 2edgeconnected graphs, critically 2connected graphs, critically 2edgeconnected graphs, 3edgeconnected graphs, 2connected cubic graphs and 3connected cubic graphs. We also give a characterization of minimally 3connected graphs.
History and Progress of the Generation of Structural Formulae in Chemistry and its Applications
"... After a few remarks on the history of molecular modelling we describe certain mathematical aspects of the generation of molecular structural formulae. The focus is on the automatic generation of structural formulae for the purpose of molecular structure elucidation and the examination of molecular l ..."
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After a few remarks on the history of molecular modelling we describe certain mathematical aspects of the generation of molecular structural formulae. The focus is on the automatic generation of structural formulae for the purpose of molecular structure elucidation and the examination of molecular libraries. The aim is to give a review and to point to relevant literature. We demonstrate an application in the area of quantitative structureproperty/activity relationships. Then, we give a glance on ongoing research in the generation of 3Dstructures (stereoisomers and conformers), and finally we mention two problems that should be solved in the near future, the possible use of hypergraphs, and the generation of patent libraries.