= Counting finite group actions by transitive orbit types
{title2=$\prod_{[H]}(1-t^{[G:H]})^{-1}$}
For a finite <group> $G$, transitive finite <group actions> are classified up to equivalence by conjugacy classes of <subgroups>: the class of $H$ gives the <coset action> on $G/H$. Every finite action decomposes uniquely as a disjoint union of its <orbits>, so its equivalence type is a multiset of these transitive types. Consequently the coefficient of $t^n$ in the displayed product, with one factor for every subgroup conjugacy class, counts actions on $n$ points up to relabeling. Each factor expands as a <geometric series> recording any multiplicity of that orbit type. For both $C_6$ and $S_3$ there is exactly one transitive type of each size $1,2,3,6$; their coefficient at degree six is $4+2+1+1=8$.
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