In this research and review paper, using some inequalities and relations
involving e, we count the number of paths and cycles in complete graphs the
number of derangements. Connection by e yields some nice formulas for the
number of derangements, and using these relations allow us to compute some incomplete gamma functions
and hypergeometric summations; these connections are hidden in the heart of
a nice polynomial that we call it derangement function and a simple ordinary
differential equation concerning it.