We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by \QSCGZ" and Guerin (unpublished, 2001).We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups.

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Conditions for transmission of a pi-conjugated molecular conductor are derived within the source and sink potential approach in terms of numbers of nonbonding levels of four graphs: The molecular graph G and the three vertex-deleted subgraphs obtained by removing one or both contact vertices. For all bipartite and most nonbipartite G, counting nonbonding levels gives a simple necessary and sufficient condition for conduction at the Fermi level. The exceptional case is where G is nonbipartite and all four graphs have the same number of nonbonding levels; then, an auxiliary requirement involving tail coefficients of the four characteristic polynomials must also be checked.

Conjugated-circuit models for induced π ring currents differ in the types of circuit that they include and the weights attached to them. Choice of circuits for general π systems can be expressed compactly in terms of matchings of the circuit-deleted molecular graph. Variants of the conjugated-circuit model for induced π currents are shown to have simple closed-form solutions for linear polyacenes. Despite differing assumptions about the effect of cycle area, all the models predict the most intense perimeter current in the central rings, in general agreement with ab initio current-density maps. All tend to overestimate the rate of increase with N of the central ring current for the [N]polyacene, in comparison with molecular-orbital treatments using ipsocentric ab initio, pseudo-π, and Hückel-London approaches.

A ranking function for the permutations on n symbols assigns a unique integer in the range [0, n! − 1] to each of the n! permutations. The corresponding unranking function is the inverse: given an integer between 0 and n! − 1, the value of the function is the permutation having this rank. We present simple ranking and unranking algorithms for permutations that can be computed using O(n) arithmetic operations.  2001 Elsevier Science B.V. All rights reserved.Keywords: Permutation; Ranking; Unranking; Algorithms; Combinatorial problems A permutation of order n is an arrangement of n symbols. For convenience when applying modular arithmetic, this paper considers permutations of {0, 1, 2, . . ., n − 1}. The set of all permutations over {0, 1, 2, . . ., n − 1} is denoted by S n .There are many applications that call for an array indexed by the permutations in S n [2]. One example is the development of programs that search for Hamilton cycles in particular types of Cayley graphs [10,11]. To do such indexing, what is desired is a bijective ranking function r that takes as input a permutation π and produces r(π), a number in the range 0, 1, . . ., n! − 1. The inverse of r is also often useful, and is called the unranking function.The traditional approach to this problem is to first define an ordering of permutations and then find ranking and unranking functions relative to that ordering.