Group action, Polya-Redfield-De Bruijn Counting and its application

在这个研究项目中，我们重新考虑了Polya-Redfield-de Bruijn关于给定置换群的非等价模式计数的经典母函数公式，重点是权值。给出了十二重生成函数的计算公式。此外，我们追求一个权重的可能性，它逐个给出所有的模式。另一方面，我们将Cayley树的Prufer码推广到1-连通但不是2-连通的连通标号图上，作为对第12行的计数的一个应用。这是在我们数学物理研究人员的建议下完成的，并打算用图论术语来理解统计力学中非理想气体状态方程的簇展开式中系数的组合意义。通过这些，我们主要打算在组合计数、P61Ya的计数理论和母函数等代数组合学方面做出贡献。所取得的成果已在两次国际研究会议上公布，部分成果正在出版中。尽管整个研究存在许多不尽如人意的地方，特别是在逐一列举的情况下，我们还是在这一科学研究的辅助下，报告了我们在这一学期的研究成果。我们感谢许多匿名合作者的帮助。

In this research project we reconsider the classical generating function formulae of enumeration of non-equivalent patterns with respect to given permutation group, due to Polya-Redfield-De Bruijn, with focus to the weight. We give a formula of generating function of the Twelvefold Way. Furthermore we pursue possibility of a weight which gives all the patterns one by one. On the other hand we extend the Prufer code of Cayley trees to connected labeled graphs those are 1-connected but not 2-connected, as an application of counting of the 3^<rd> row of the Twelvefold Way. This is done after an advice of our mathematical physics investigator, and intend to understand combinatorial meaning of a coefficients in the cluster expansion of state equation of an imperfect gas in statistical mechanics, by graph theory terms. Through these we mainly intend to contribute to algebraic combinatorics, such as combinatorial enumeration, P61Ya's counting theory and generating function.Obtained results were announced by 2 international research conferences, and part of those are in press now. Though there are many unsatisfactory points as whole of the research, especially in one by one enumeration, we give a report of our research results in this term of project, with this grand-in-aid for scientific research. We appreciate helps of many anonymous cooperators.