Enumeration and Classification of Certain Polygonal Systems Representing Polycyclic Conjugated Hydrocarbons: Annelated Catafusenes

Abstract

Catafusenes (catacondensed simply connected polyhexes) annelated to a core are studied. Schemes for handling the enumeration of these systems with four appendages or less are given. In this survey the formalism of combinatorial summations is employed, but the exposition is supported by applications of generating functions. Special cases of interest occur when the core is a g-gon (g-membered polygon). Then the systems are referred to as catacondensed mono-g-fusenes, which are subclasses of the important mono-g-polyhexes.

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

“…However, a further application of the triangular A matrix seems to be more interesting. It was mentioned that A has been used in the enumeration of di-4-catafusenes, but a generalization to α-4-catafusenes is undoubtedly feasible. An α-4-catafusene consists of α tetragons and otherwise only hexagons (if any).…”

“…The enumeration method called “stupid sheep counting” 17 is very useful 18,19 and has been exploited in the present treatment of unbranched α-5-catafusenes . In this case, the “crude total”, ,, an important concept in stupid sheep countings, counts the D 2 h systems exactly once, the C 2 h and C 2 v systems twice each, and the C s systems four times. For a given number of polygons ( r ), the numbers of the systems with the different symmetries are denoted by D r , C r , M r and A r , pertaining to the symmetry groups in the same order as mentioned above.…”

“…The unbranched mono-5-catafusene systems consist of unbranched catafusenes attached to a pentagon. The enumeration problem of catafusenes rooted at a polygon core has been treated in general terms, , and several treatments of special cases are available. ,,, In addition, a work involving unbranched catafusenes is especially relevant.…”

“…The A matrix has been employed in the enumeration of di-4-catafusenes, which consist of two tetragons each and otherwise only hexagons. In the present studies of α-5-catafusenes, the matrix is of prime interest.

3 The unbranched mono-5-catahelicenes with r = 6, 7, and the C 2 v such systems with r = 9.

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

“…They contain mono- q -polyhexes, − which are systems containing exactly one q -gon each and otherwise only hexagons. Works on different systems of this category have been reported several times in this journal ,,− and elsewhere; − , the reader is also referred to the bibliography in one of the cited papers …”