“Subfactorial”的概念、定义、习题解题方法及翻译-数学辞典

单词

Subfactorial

释义

Subfactorial

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

(1)

(2)

where

is the usual Factorial and

is the Nint function. The first few values are

, ... (Sloane's A000166). For example, the onlyDerangements of

are

and

, so

. Similarly, theDerangements of

are

, and

, so

The subfactorials are also called the Rencontres Numbers and satisfy the RecurrenceRelations

(3)

(4)

The subfactorial can be considered a special case of a restricted Rooks Problem.

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

(5)

(Madachy 1979).

See also Derangement, Factorial, Married Couples Problem, Rooks Problem, Superfactorial

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. SequenceA000166/M1937in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and extended entry inSloane, 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.

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