Computational generating function techniques are outlined for combinatorics of colorings of all hyperplanes of the 6D-hypercube for 65 irreducible representations of the 6D-hyperoctahedral group isomorphic to the wreath product S 6 [S 2 ] group of order 46,080. The computational techniques are inspired by a number of physico-chemical and biological applications to molecular chirality, molecular clusters, isomerization reaction graphs, relativistic effects, massively-large data sets, visualization, and genetic regulatory networks. Computational techniques are comprised of computing the generalized character cycle indices of 65 irreducible representations for all hyperplanes of the 6D-hypercube using the Möbius inversion technique followed by the construction of polynomial generators for different cycle types under the hyperoctahedral group action for all six types of hyperplanes of the 6D-hypercube. Subsequently, multinomial generating functions for colorings of all (6-q)-hyperplanes of the 6D-hypercube are constructed for q = 1 through 6. We have presented tables thus computed for the combinatorics for colorings of six hyperplanes of 6D-hypercube for 65 irreducible representations and outline applications to chemical and biological sciences.

中文翻译：

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用于所有不可约表示和应用的 6D 超立方体超平面着色的计算组合

计算生成函数技术概述了 6D 超立方体的所有超平面的着色组合，用于同构于 46,080 阶的花环积 S 6 [S 2 ] 群的 6D 超八面体群的 65 个不可约表示。计算技术的灵感来自于分子手性、分子簇、异构化反应图、相对论效应、大规模数据集、可视化和遗传调控网络的许多物理化学和生物学应用。计算技术包括使用莫比乌斯反演技术为 6D 超立方体的所有超平面计算 65 个不可约表示的广义特征循环索引，然后在所有六种超平面的超八面体群作用下为不同循环类型构建多项式生成器6D 超立方体。随后，为 q = 1 到 6 构造了用于 6D 超立方体的所有 (6-q)-超平面着色的多项式生成函数。我们已经提出了这样计算的表，用于对 65 的 6D 超立方体的六个超平面的着色进行组合不可约表示和概述在化学和生物科学中的应用。