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

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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.