Chemical Combinatorics for Alkane-Isomer Enumeration and More L. Bytautas ² and D. J. Klein* Texas A&M University at Galveston, Galveston, Texas 77553-1675 Received May 18, 1998 Standard combinatorial enumeration techniques for alkanes are considered with a view to the extension to a widened range of chemically interesting features. As one brief point it is noted that these standard techniques naturally associate to generational schemes and thence have nomenclatural interpretations, which may be made to achieve some similarity to the standard IUPAC nomenclature. Our primary focus is the illustration that such combinatorial techniques are sufficient to enable computation of several graph-theoretic structural invariants averaged over (different types of) isomer classes. Such averages (and associated isomer counts) are tabulated for structural isomers for up to N ) 40 carbons, where there are ∼10 14 isomers (though the computational methodology should rather readily extend to at least N ) 80 where there should be ∼10 28 isomers). The averages for invariants are utilized to estimate several physicochemical properties averaged over these same isomer classes. The properties currently so considered are heat of formation, index of refraction, and magnetic susceptibility. Further, various asymptotic results for counts, mean invariants, and mean properties are noted, so that the exact graph-theoretic data are extrapolated with high accuracy to arbitrarily large alkanes. 1. INTRODUCTION One of the classic areas of chemistry concerns isomerism, starting in the 1860s, when it was realized (as in ref 1) that there were different possible structural formulas for the same atomic composition. That such different structural formulas corresponded to different compounds provided crucial evi- dence for the validity of classical structural formulas and the existence of structured molecules. The problem of formal isomer enumeration was initiated by Cayley 2 in 1874 for the case of alkanes, and this continued as a topic of interest for some time, 3 particularly with the group of Henze 4 making several enumerations, for alkanes and homologous sequences of various derivatives. Then in 1935 motivated by the chemical isomer problem Po ´lya 5,6 developed a powerful combinatorial theory for the enumeration of symmetry- mediated equivalence classes of “colorings”, and this enu- meration theory has now become standard fare in combina- torics texts. Beyond the formal mathematical theory, Po ´lya applied his theory to a few chemical problems, but in particular the problem of alkane isomer enumeration, and following Po ´lya there have been further (often very formal) refinements for this chemical problem in numerous papers, e.g., as refs 7-16. Work has been done emphasizing 7,8,10,13 the generality of the method in dealing with additional systems beyond alkanes, and there has been (a lesser amount of) work 5,9-12 dealing with the asymptotic behavior of the isomer counts, primarily for alkanes. The pre-1986 work on Po ´lya enumeration for chemical purposes is nicely reviewed by Read 7 who also gives a translation (made by D. Aeppli) of Po ´lya’s foundational paper, and Fujita 13 considers in detail many recent theoretical extensions (primarily concerning chirality and symmetry questions not only for isomers but also for reaction processes). Yeh 17 and Cyvin and co-workers 18,19 have recently repeatedly noted that much of Po ´lya’s general mathematical formalism can be foregone in many specific isomer enumerationssthat is, whole chapters often occurring in standard combinatorics texts concerning formal consideration of permutation groups, general “cycle index” functions, and “pattern inventories” as well as general theory of implicit and holomorphic functions often need not be gone through for many specific cases. Here we adopt this straightforward point of view for the classic case of alkanes, thereby refining some earlier results, and we develop possibilities for some extensions to new chemically oriented applications beyond simple isomer enumeration. Initially in section 2 the classical enumeration problem is reconsidered using such a straightforward ap- proach. It is noted in passing that there is a potential relation of these mathematical enumeration procedures to structure generation and associated nomenclatural systems, and this is detailed in the Appendix. The primary aim here of developing a more widely useful combinatoric chemistry is pursued in sections 3 and 4 where we show how the standard combinatorial enumerative techniques can be readily ex- tended to compute average values of certain graph-theoretic invariants, the averages being over the members of isomer classes. A few averages of some interest in “polymer statistics” are considered, including average numbers of conformations per isomer, the average through-bond diameter of a class of isomers, and the average through-bond (or shortest-path) distance between carbons in a class of isomers. This last quantity is obtained via a computation of the average so-called 20,21 “Wiener number” for isomeric alkanes. Also average counts of primary, secondary, tertiary, and quaternary carbons are made. Such graph-theoretic invari- ² On leave from Institute of Theoretical Physics & Astronomy, Gostauto 12, 2600 Vilnius, Lithuania. 1063 J. Chem. Inf. Comput. Sci. 1998, 38, 1063-1078 10.1021/ci980095c CCC: $15.00 © 1998 American Chemical Society Published on Web 10/17/1998