Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile

The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number $φ_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ 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 $φ_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $φ_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.

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