Let $>$ be a linear order relation over a set $A$. Consider the graph $G$ that represent the transitive closure of $>$. Obviously $G$ is directed and acyclic. Given a set of edges $(e_1,e_2,\dots,e_n)$ from $G$, What is the number of linear orders that agrees with the edges?
A linear order $>'$ agrees with edge $e_i=(x,y)$ if $x>' y\iff x>y$.