Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

We present a complete set of closed-form formulas for the ZZ polynomials of five classes of composite Kekulean benzenoids that can be obtained by overlapping two parallelograms: generalized ribbons Rb, parallelograms M, vertically overlapping parallelograms MvM, horizontally overlapping parallelograms MhM, and intersecting parallelograms MxM. All formulas have the form of multiple sums over binomial coefficients. Three of the formulas are given with a proof based on the interface theory of benzenoids, while the remaining two formulas are presented as conjectures verified via extensive numerical tests. Both of the conjectured formulas have the form of a 2x2 determinant bearing close structural resemblance to analogous formulas for the number of Kekule structures derived from the John-Sachs theory of Kekule structures.