Taylor's Law Holds for Finite OEIS Integer Sequences and Binomial Coefficients

Taylor's law (TL) predicts that the variance and the mean will be related empirically through a power-law function. TL previously has been shown to arise even in the absence of biological, ecological or physical processes. We report here that the mean and variance of 110 finite integer sequences in the On-Line Encyclopedia of Integer Sequences (OEIS) obey TL approximately. We also show that the binomial coefficients on each row of Pascal's triangle obey TL asymptotically. These applications of TL to seemingly unrelated mathematical structures tend to confirm there might be purely statistical, context-independent mechanisms at play. Supplementary materials for this article are available online.

泰勒定律（Taylor's Law, TL）预测，方差与均值将通过幂律函数在经验层面建立关联。此前研究表明，即便不存在生物学、生态学或物理学过程，泰勒定律依然可以成立。本文报告，整数序列在线百科全书（On-Line Encyclopedia of Integer Sequences, OEIS）中收录的110条有限整数序列的均值与方差近似符合泰勒定律。此外，本文还证明，帕斯卡三角形（Pascal's triangle）每一行的二项式系数渐近符合泰勒定律。将泰勒定律应用于这些看似无关的数学结构，结果倾向于佐证：或许存在纯粹统计学层面、与场景无关的底层机制在发挥作用。本文的补充材料可在线获取。