  We present a structural approach of some results about jumps in the behavior
of the profile (alias generating function) of hereditary classes of finite
structures. We consider the following notion due to N.Thi\'ery and the second
author. A \emph{monomorphic decomposition} of a relational structure $R$ is a
partition of its domain $V(R)$ into a family of sets $(V_x)_{x\in X}$ such that
the restrictions of $R$ to two finite subsets $A$ and $A'$ of $V(R)$ are
isomorphic provided that the traces $A\cap V_x$ and $A'\cap V_x$ have the same
size for each $x\in X$. Let $\mathscr S_\mu $ be the class of relational
structures of signature $\mu$ which do not have a finite monomorphic
decomposition. We show that if a hereditary subclass $\mathscr D$ of $\mathscr
S_\mu $ is made of ordered relational structures then it contains a finite
subset $\mathfrak A$ such that every member of $\mathscr D$ embeds some member
of $\mathfrak A$. Furthermore, for each $R\in \mathfrak A$ the profile of the
age $\age(R)$ of $R$ (made of finite substructures of $R$) is at least
exponential. We deduce that if the profile of a hereditary class of finite
ordered structures is not bounded above by a polynomial then it is at least
exponential. For ordered graphs, this result is a part of classification
obtained by Balogh, Bollob\'as and Morris (2006).
