烃同分异构体的计数

Isomer Enumeration of Simple Hydrocarbons

能否系统精确地解决高中化学学习中的同分异构体计数问题?本文在高中数学化学的背景知识之上,引入生成函数以及Pólya计数定理,在不考虑立体异构的前提下,从烷基的同分异构体计数入手,逐步推导烷烃、一烯烃、一炔烃苯的衍生物、二取代烷烃以及二烯烃的同分异构体计数的公式,并将计算结果与手工枚举进行对比,进一步证明其正确性。我们还归纳性地发掘了同分异构体计数的渐进特征并提出了以上烃类同分异构体计数的近似公式。对比国外的成果,本文尽力避免引入过多的数学概念,降低门槛,适合高中生阅读,加深对于烃结构的理解,培养计算思维;对比国内相同主题的研究,本文的推导形式清晰简洁,拓展性较强。

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.