The Polya Theory and Permutation Groups
This paper presents a thorough exposition of the Polya Theory in its enumerative applications to permutations groups. The discussion includes the notion of the power group, the Burnside’s Lemma along with the notions on groups, stabilizer, orbits and other group theoretic terminologies which are so fundamentally used for a good introduction to the Polya Theory. These in turn, involve the introductory concepts on weights, patterns, ﬁgure and conﬁguration counting series along with the extensive discussion of the cycle indexes associated with the permutation group at hand. In order to realize the applications of the Polya Theory, the paper shows that the special ﬁgure series $c(x) = 1 + x$ is useful to enumerate the number of $G$-orbits of $r$-subsets of any arbitrary set $X$. Furthermore, the paper also shows that this special ﬁgure series simpliﬁes the counting of the number of orbits determined by any permutation group which consequently determines whether or not the said permutation group is transitive.