This question is meant to be applied to recover historic information from genetic data. The following model, is (probably) the simplest possible which takes recombinations into account.
First, let us introduce some terms:
human is a finite set of numbers in (0,1); further these numbers will be called "scars".
population is a finite set of humans
One can perform one operation with a population:
- recombination; i.e. take two humans $A$ and $B$ from the population, choose randomly a number $x\in(0,1)$ and produce a new human $C$ which has scar $x$, all scars in $A$ which are $ < x$ and all scars in $B$ which are $ > x$.
Assume someone starts with a population formed by two empty sets and does these operations for a while then stop. Assume you have all information about a portion of the population, BUT you do not know how this portion had been chousen. What one can say (even in which terms) about the history of population.
It is not exactly mathematical problem, an answer might be something like "colored graph".
Clearly one can not say everything; yet there are many questions which can be answered. For example, assume you want to know if it is likely that at some moment your population was divided in two and there were no interbreeding between these groups for a while. You can even estimate "time" when they split. BUT I'm interested in a more abstract way to describe the history --- I want to say something without any assumptions.
Something is possible: First note that with probability 1, there is one-to-one correspondence between humans and scars. Given scar $x$, let us denote by $H_x$ the corresponding human. Assume in the portion of population you have two humans: human $A$ who has scars $x < y$ and nothing in between $x$ and $y$ and human $B$ who has scars $x < z$ and nothing in between. Assume $z < y$ then $H_z$ is a descendant of $H_x$. That gives a partial order on all such humans. Is it all one can do?