Enumeration of Poly-5-catafusenes Representing a Class of Catacondensed Polycyclic Conjugated Hydrocarbons

Abstract

This work deals with isomer enumerations of α-5-catafusenes. An α-5-catafusene is a catacondensed polygonal system consisting of α pentagons and otherwise only hexagons (if any). The numbers of isomers of mono-5-catafusenes and tri-5-catafusenes from computer programming are reported. For the Ir numbers of unbranched α-5-catafusenes, where r is the number of polygons, a complete mathematical solution is described. A new approach is employed, which involves a triangular matrix with interesting mathematical properties. The ultimate result is an explicit formula for Ir in r and α.

Enumeration of Unbranched Catacondensed Polygonal Systems: General Solution for Two Kinds of Polygons

“…For the sake of brevity, we shall not go into detail of the combinatorial reasonings behind the deductions. Instead, the reader is referred to previous treatments of different special cases. − In addition, the present deductions are supported by numerical examples. For this purpose, the unbranched α-6-catapolyheptagons ( p = 6, q = 7) were selected.…”

“…The matrix itself is triangular and is denoted A ( x , y ). Then the matrices A and which were introduced in a previous paper, are the special cases A (2,1) and A (1,2), respectively. Furthermore, A (1,1) is the Pascal triangle Another example is Notice that the first row of nonvanishing elements in an A ( x , y ) matrix is always [1], while the second row is [ x y ].…”

“…Firstly, the enumerations of unbranched di-4-catafusenes were generalized to unbranched α-4-catafusenes (α tetragons, r − α hexagons) . Secondly, the corresponding problem was solved for unbranched α-5-catafusenes (α pentagons, r − α hexagons) …”

“…But this is exactly what we have achieved in the present work; a complete mathematical solution was derived for the numbers of unbranched catacondensed polygonal systems with two kinds of polygons of arbitrary sizes. This is a generalization of which several special cases (including those with one kind of polygons) have been studied by different investigators. ,,− The problem turned out to be surprisingly complicated; nevertheless, it was solved in terms of explicit formulas. Presumably, the present work represents the highest degree of generalization which practically can be achieved in closed form.…”

Fixed Bonds in a Class of Polygonal Systems

“…The first syxtematic studies of indacenoids are due to Dias (3,4). There are many results about indacenoids (5)(6)(7)(8)(9). So far, it is known that each of the catacondensed indacenoids has at least one fixed single bond, but none of them has fixed double bonds (10).…”