  Let $G$ be a group of permutations of a denumerable set $E$. The profile of
$G$ is the function $\phi_G$ which counts, for each $n$, the (possibly
infinite) number $\phi_G(n)$ of orbits of $G$ acting on the $n$-subsets of $E$.
Counting functions arising this way, and their associated generating series,
form a rich yet apparently strongly constrained class. In particular, Cameron
conjectured in the late seventies that, whenever $\phi_G(n)$ is bounded by a
polynomial, it is asymptotically equivalent to a polynomial. In 1985,
Macpherson further asked if the orbit algebra of $G$ - a graded commutative
algebra invented by Cameron and whose Hilbert function is $\phi_G$ - is
finitely generated. In this paper, we announce a proof of a stronger statement:
the orbit algebra is Cohen-Macaulay. The generating series of the profile is a
rational fraction whose numerator has positive coefficients and denominator
admits a combinatorial description. The proof uses classical techniques from
group actions, commutative algebra, and invariant theory; it steps towards a
classification of ages of permutation groups with profile bounded by a
polynomial.
