We present here a survey of most notable Delannoy's works. These works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the rst general way to solve Ballot-like problems. We also give a tentative short biography.

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.

Recursive equations are derived for the exact number t h of nonisomorphic free trees which have some rooting as a binary tree of height h. Numerical results are calculated using these formulae. 1. Introduction A binary tree T can be defined as a rooted tree in which each node has degree at most 3, except that the root has degree at most 2. The height of T is the maximum distance from the root node to an endnode. Binary trees are much used in theoretical computer science, with height often being a key parameter directly related to the efficiency of associated algorithms. A free binary tree F is an unrooted tree which has a node u (not necessarily unique) such that F is a binary tree when rooted at u. Our purpose is to derive formulae for the number of unlabeled free binary trees which have a rooting that produces a binary tree of height h; we say that such a tree admits height h. In general our terminology follows [3]. Unlabeled counting does not distinguish between versions of a tree...

Generalized recoupling coefficients or 3nj-coefficients for a Lie algebra (with su(2), the Lie algebra for the quantum theory of angular momentum, as generic example) can always be expressed as multiple sums over products of Racah coefficients (i.e. 6j-coefficients). In general there exist many such expressions; we say that such an expression is optimal if the number of Racah coefficients in such a product (and, correlated, the number of summation indices) is minimal. The problem of finding an optimal expression for a given 3nj-coefficient is equivalent to finding a shortest path in a graph Gn . The vertices of this graph Gn consist of binary coupling trees, representing the coupling schemes in the bra/kets of the 3nj-coefficients. This is the graph of rooted (unordered) binary trees with labelled leaves, and has order (2n \Gamma 1)!!. As the order increases so rapidly, finding a shortest path is computationally achievable only for n ! 11. We present some mathematical tools ...

This paper considers the problem of enumerating all non-isomorphic tree-like chemical graphs with given path frequency, where “tree-like ” means that the graph can be viewed as a tree if multiple edges (i.e., edges with the same end points) and a benzene ring are treated as one edge and one vertex, respectively, and “path frequency ” is a vector of the numbers of specified vertex-labeled paths that must be realized in every output. This and related problems have several potential applications such as classification of chemical compounds, structure determination using mass-spectrum and/or NMR and design of novel chemical compounds. For this problem, several studies have been done. Recently, Fujiwara et al. (2008) showed two formulations and for each of them, they gave a branch-and-bound algorithm, which combined efficient enumeration of non-isomorphic trees with bounding operations based on the path frequency and the atom-atom bonds to avoid the generation of invalid trees. In this paper, based on their work and a result of Nagamochi (2006), we introduce two new bounding operations, the detachment-cut and the H-cut, to further reduce the size of the search space. We performed computational experiments to compare our proposed algorithms with those of Fujiwara et al. (2008) using some chemical compound data obtained from the KEGG LIGAND database

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Abstract. This article is not a research paper, but a little note on the history of combinatorics: We present here a tentative short biography of Henri Delannoy, and a survey of his most notable works. This answers to the question raised in the title, as these works are related to lattice paths enumeration, to the so-called Delannoy numbers, and were the first general way to solve Ballot-like problems. This version corresponds to an update (May 2002) of the abstract submitted (February 2002) by the first author to the 5th lattice path combinatorics and discrete distributions