What is the probability of having no cycles with length $n$ (touching all vertices) in a simple, directed Erdos-Renyi graph with $n$ vertices?
For example, if $n=2$, then the probability is $1-P(cycle)=1-p^2$. If $n=3$, then the probability is $1-P(cycle)=1-(p^3+p^3-p^6)$.