Counting and Computing by $e$

Abstract

In this paper we count the number of paths and cycles in complete graphs by using the number $e$. Also, we compute the number of derangements in same way. Connection by $e$ yields some nice formulas for the number of derangements, such as $D_n=\lfloor \frac{n!+1}{e}\rfloor$ and $D_n=\lfloor (e+e^{-1})n!\rfloor -\lfloor en!\rfloor$, 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.

12 pages, no figure, review of my works about the number of derangements