Journal of Integer Sequences, Vol. 23 (2020), Article 20.7.8

Derangements and Alternating Sum of Permutations by Integration

Mehdi Hassani
Department of Mathematics
University of Zanjan
University Blvd.
45371-38791 Zanjan


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 Dnn!/e, where Dn 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 Dn with coefficients in terms of the Bell numbers Bn. 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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