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BUILT-IN MATHEMATICA SYMBOL

Subfactorial

Subfactorial[n]
gives the number of permutations of n objects that leave no object fixed.

For non-integer n, the numerical value of Subfactorial[n] is given by Gamma[n+1, -1]/E.

A permutation in which no object appears in its natural place is called a derangement.

Out[1]=

Evaluate for large numbers:

Evaluate numerically for non-integer arguments:

Evaluate for complex arguments:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Subfactorial automatically threads over lists:

There are 9 derangements of a set of 4 objects:

Here are all permutations of the set

Delete all permutations where an object is fixed:

Check that there are only 9 derangements:

Subfactorial[n] is given by

Recurrence relations satisfied by Subfactorial:

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