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Derangements and Alternating Sum of Permutations by Integration
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Mehdi Hassani

Department of Mathematics

University of Zanjan

University Blvd.

45371-38791 Zanjan

Iran

**Abstract:**

Let *P*(*n*,*j*) denote the number of *j*-permutations of
*n* objects. In
this paper we obtain
the generating function for the alternating sequence
(-1)^{j} *P*(*n*,*j*).
Our method gives an integral representation
for the difference *D*_{n} – *n*!/*e*,
where *D*_{n} denotes the number of
derangements on *n* objects.
Using this integral representation, we compute
the moments of this difference, and we also get an asymptotic expansion
for *D*_{n} with coefficients in terms of
the Bell numbers *B*_{n}.
We also give a simple proof of the irrationality of *e*.

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(Concerned with sequences
A000110
A000166.)

Received March 23 2020; revised version received July 28 2020.
Published in *Journal of Integer Sequences*,
July 29 2020.

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