# How many topological orderings exist for a graph?

Given a directed and acyclic graph G(V,E) is there any way to determine the exact number of topological orderings it can have? I know the minimum is 1 if it contains a Hamiltonian path, but can the maximum number be determined ?

• Assuming that the graph is directed and acyclic (which is has to be for a topological ordering to make sense), if the graph has a Hamiltonian path then there is a $\mathit{unique}$ topological sort ordering, not just a minimum of $1$. – Peter Woolfitt May 29 '14 at 20:33