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Pólya theory for species with an equivariant group action

Joyal's theory of combiantorial species provides a rich and elegant framework for enumerating combinatorial structures by translating structural information into algebraic functional equations. We present some classical and folklore results which interpret the species-theoretic cycle index series in terms of the Pólya theory of the action of the symmetric group on the label set, allowing the enumeration of "partially-labeled" structures and providing an alternate foundation for several proofs. We also extend the theory to incorporate information about "structural" group actions (i.e. those which commute with the label permutation action) on combinatorial species, using the $Γ$-species of Henderson, and present Pólya-theoretic interpretations of the associated formal power series. We define the appropriate operations $+$, $\cdot$, $\circ$, and $\square$ on $Γ$-species, give formulas for the associated operations on $Γ$-cycle indices, and illustrate the use of this theory to study several important examples of combinatorial structures. Finally, we demonstrate the use of the Sage computer algebra system to enumerate $Γ$-species and their quotients.