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  "body": "\n## Proof\n\nIt follows from the re-labelling construction that there are\n\n$$\\frac{1}{|G|} \\sum_{r_1 + r_2 + \\cdots + r_n = k} {k \\choose r_1, r_2, \\ldots, r_n} \\; r_1! [z^{r_1}] g(z) \\; r_2! [z^{r_2}] g(z) \\; \\cdots \\; r_n! [z^{r_n}] g(z)$$\n\nor\n\n$$\\frac{k!}{|G|} \\sum_{r_1 + r_2 + \\cdots + r_n = k} [z^{r_1}] g(z) [z^{r_2}] g(z) \\cdots [z^{r_n}] g(z) \\; = \\; \\frac{k!}{|G|} [z^k] g(z)^n$$\n\ndifferent configurations of total size *k*. The formula evaluates to an integer because $[z^k] g(z)^n$ is zero for *k* \\\u003c *n* (remember that *g* does not include objects of size zero) and when $k \\ge n$ we have $n!|k!$ and the order $|G|$ of *G* divides the order of $S_n$, which is $n!$, by [[Lagrange's theorem|3C53TD]]. The conclusion is that the EGF of the labelled configurations is given by\n\n$$f_n(z) = \\sum_{k \\ge 0} \\left( \\frac{k!}{|G|} [z^k] g(z)^n \\right) \\frac{z^k}{k!} = \\frac{1}{|G|} \\sum_{k \\ge 0} z^k [z^k] g(z)^n = \\frac{g(z)^n}{|G|}.$$\n\nThis formula could also be obtained by enumerating sequences, i.e. the case when the slots are not being permuted, and by using the above argument without the $1/|G|$-factor to show that their generating function under re-labelling is given by $g(z)^n$. Finally note that every sequence belongs to an orbit of size $|G|$, hence the generating function of the orbits is given by $g(z)^n/|G|.$\n",
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