关键词: 烃, 同分异构体, 计数, Pólya计数定理

Abstract:

Can we count hydrocarbon isomers systematically and accurately? This article demonstrates a neat solution by introducing generating functions and Pólya's Counting Theorem based on high-school-level mathematics and chemistry knowledge. Starting with alkyls, we derive formulae to enumerate acyclic isomers (stereoisomerism not considered) of alkanes, monoalkenes, monoalkynes, benzene derivatives, disubstituted alkanes and dienes. We then check our results against manual enumeration to show their correctness. We also explore the asymptotic behavior of isomerism and empirically propose approximation formulae for counting isomers of the aforementioned hydrocarbons. Compared to results from literature, we strive to reduce advanced mathematical concepts involved to eliminate barriers for inquisitive high school readers, who may gain a more in-depth understanding of hydrocarbon structure and train their computational thinking. Compared to results from Chinese literature, our derivation is more concise, flexible and extendable.

Key words: Hydrocarbon, Isomer, Enumeration, Pólya's Counting Theorem