Profile and hereditary classes of ordered relational structures

Let $\mathfrak{C}$ be a class of finite combinatorial structures. The $\mathit{\text{profile}}$ of $\mathfrak{C}$ is the function ${\phi }_{\mathfrak{C}}$ which counts, for every integer $n$, the number ${\phi }_{\mathfrak{C}}(n)$ of members of $\mathfrak{C}$ defined on $n$ elements, isomorphic structures been identified. The $\mathit{\text{generating}}\mathit{\text{function}}\mathit{\text{of}}$ $\mathfrak{C}$ is $\mathcal {H}{\mathfrak{C}}(x):=\sum{n\geqq 0}\varphi_{\mathfrak{C}}(n)x^{n}$. Many results about the behavior of the function ${\phi }_{\mathfrak{C}}$ have been obtained. Albert and Atkinson have shown that the generating series of several classes of permutations are algebraic. In this paper, we show how their results extend to classes of ordered binary relational structures; putting emphasis on the notion of hereditary well quasi order, we discuss some of their questions and answer one.