Counting and Computing by $e$

Details Counting and Computing by $e$ Counting and Computing by $e$

2006-06-26

arxiv.org/abs/math/0606613v1

M. Hassani, Derangements and Applications, Journal of Integer Sequences (JIS), Volume 6, Issue 1, Article 03.1.2, 2003. M. Has

Mathematics

Combinatorics

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

Scientific paper

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.

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