Methods of enumerating Kekulé structures, exemplified by applications to rectangle-shaped benzenoids

Abstract

The enumeration of Kekulé structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekulé structure count) are derived by several methods. The oblate rectangles, R^j(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For R^j(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the “perforated” oblate rectangles, is considered in order to exemplify a “perfectly explicit” combinatorial K formula, an expression for arbitraty values of the parameters m and n.