权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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

群体行动、Polya-Redfield-De Bruijn 计数及其应用

基本信息

批准号：
13640081
负责人：
KAJIMOTO Hiroshi
金额：
$ 2.11万
依托单位：
Nagasaki University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640081/
关键词：
Prufer code counting blocks Polya theory group action generating function words polya theory Prtifer code biconnected component generationg function

项目摘要

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.

在本研究项目中，由于Polya-Redfield-De Bruijn的原因，我们重新考虑了关于给定置换群的非等价模式枚举的经典生成函数公式，重点是权重。给出了十二重道生成函数的一个公式。此外，我们追求一个权重的可能性，它一个接一个地给出所有的模式。另一方面，我们将Cayley树的Prufer码扩展到1连通而非2连通的连通标记图，作为十二重道3^<rd>行计数的应用。这是在我们的数学物理研究者的建议下完成的，并打算用图论的术语来理解统计力学中不完全气体状态方程的簇展开中系数的组合意义。通过这些，我们主要致力于代数组合学，如组合枚举，P61Ya的计数理论和生成函数。获得的结果已在2个国际研究会议上公布，部分成果正在出版中。虽然整个研究中有很多不尽人意的地方，尤其是在一一列举的时候，但是我们还是把这个学期的研究成果做了一个报告，作为对科学研究的一种大力资助。我们感谢许多匿名合作者的帮助。