Some relational structures with polynomial growth and their associated algebras II

Details Some relational structures with polynomial growth and their associated algebras II Some relational structures with polynomial growth and their associated algebras II

2008-01-29

arxiv.org/abs/0801.4404v1

Mathematics

Combinatorics

10 pages, submitted to ROGICS'08

Scientific paper

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.

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