The number of Permutations of objects in which no object appear in its natural place (i.e.,
so-called ``Derangements'').

(1) |

(2) |

The subfactorials are also called the Rencontres Numbers and satisfy the Recurrence
Relations

(3) |

(4) |

The only number equal to the sum of subfactorials of its digits is

(5) |

**References**

Dörrie, H. §6 in *100 Great Problems of Elementary Mathematics: Their History and Solutions.*
New York: Dover, pp. 19-21, 1965.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, p. 167, 1979.

Sloane, N. J. A. Sequence
A000166/M1937
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in
Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Stanley, R. P. *Enumerative Combinatorics, Vol. 1.* Cambridge, England: Cambridge University Press, p. 67, 1997.

© 1996-9

1999-05-26