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Séminaire Lotharingien de Combinatoire, 80B.83 (2018), 12 pp.

# Justine Falque and Nicolas M. Thiéry

# The Orbit Algebra of an Oligomorphic Permutation Group with Polynomial Profile is Cohen-Macaulay

**Abstract.**
Let *G* be a group of permutations of a denumerable set
*E*. The *profile* of *G* is the function φ_{G} which
counts, for each *n*, the (possibly infinite) number φ_{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 φ_{G}(*n*) is bounded by a
polynomial, it is asymptotically equivalent to a polynomial. In
1985, Macpherson further asked if the \textbf{orbit
algebra} of *G* - a graded commutative algebra invented by
Cameron and whose Hilbert function is φ_{G} - is finitely
generated.
In this paper we announce a proof of a stronger statement: the orbit
algebra is Cohen Macaulay; it follows that the generating series of
the profile is a rational fraction whose denominator admits a
combinatorial description and the numerator is non-negative.

The proof uses classical techniques from actions of permutation
groups, commutative algebra, and invariant theory; it steps towards
a classification of ages of permutation groups with profile bounded
by a polynomial.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

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