Some relational structures with polynomial growth and their associated algebras II

The profile of a relational structure R is the function phi which counts for every integer n the number, possibly infinite, phi(n) of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If phi takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra, introduced by P. J. Cameron. In this paper, we give a closer look at this association, particularly when the relational structure R admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that phi is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on R, phi is eventually a quasi-polynomial whenever it is bounded by some polynomial. We also characterize when the age algebra is finitely generated.