  The cycle polynomial of a finite permutation group $G$ is the generating
function for the number of elements of $G$ with a given number of cycles:
\[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$
on $\Omega$. In the first part of the paper, we develop basic properties of
this polynomial, and give a number of examples.
  In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs
of combinatorial polynomials. We show that, in a considerable number of cases,
there is a polynomial in the reciprocal relation to the cycle polynomial of
$G$; this is the orbital chromatic polynomial of $\Gamma$ and $G$, where
$\Gamma$ is a $G$-invariant graph, introduced by the first author, Jackson and
Rudd. We pose the general problem of finding all such reciprocal pairs, and
give a number of examples and characterisations: the latter include the cases
where $\Gamma$ is a complete or null graph or a tree.
  The paper concludes with some comments on other polynomials associated with a
permutation group.
