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subfactorials
The subfactorial of an integer  , often denoted by    is equal to the number of derangements of    objects, i.e., the number of permutations with no fixed points.

For example  , because there are 9 derangments of the set  , namely  ,  ,  ,  ,  ,  ,  ,  , and  .

Three formulas for    :

where the last formula holds for  .

The first subfactorials are 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734 more terms

Subfactorials can also be... (you may click on names or numbers)