subfactorials

The subfactorial of an integer $n\ge0$ , often denoted by $!n$

is equal to the number of derangements of $n$

objects, i.e., the number of permutations with no fixed points.

For example $!4=9$ , because there are 9 derangments of the set $\{a,b,c,d\}$ , namely $\{b, a, d, c\}$ , $\{b, c, d, a\}$ , $\{b, d, a, c\}$ , $\{c, a, d, b\}$ , $\{c, d, a, b\}$ , $\{c, d, b, a\}$ , $\{d, a, b, c\}$ , $\{d, c, a, b\}$ , and $\{d, c, b, a\}$ .

Three formulas for $!n$ :

$!n = n!\sum_{i=0}^{n}\frac{(-1)^i}{i!}\, =\, \sum_{k=0}^n(-1)^{n-k}k!{n\choose k}\, =\, \left\lfloor\frac{n!+1}{e}\right\rfloor\,,$

where the last formula holds for $n\ge1$ .

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

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here

A graph displaying how many subfactorials are multiples of the primes p from 2 to 71. In black the ideal line 1/p.

Useful links Mathworld, Subfactorial
Wikipedia, Derangement
OEIS, sequence A000166

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

aban 44 265 abundant 1854 133496 alternating 1854 amenable 44 265 14833 133496 1334961 176214841 apocalyptic 14833 arithmetic 44 265 1854 14833 c.square 265 congruent 265 1854 133496 Curzon 1854 cyclic 265 14833 deficient 44 265 14833 1334961 dig.balanced 44 Duffinian 265 eban 44 economical 265 1334961 emirpimes 265 equidigital 265 1334961 evil 1854 176214841 gapful 14833 14684570 176214841 happy 44 Harshad 1854 1334961 hoax 265 iban 44 interprime 1854 133496 Lehmer 14833 magnanimous 265 modest 1854 Moran 1854 nialpdrome 44 nude 44 O'Halloran 44 odious 44 265 14833 133496 1334961 14684570 palindromic 44 pernicious 44 265 133496 1334961 plaindrome 44 practical 133496 pseudoperfect 1854 133496 repdigit 44 self 14833 14684570 semiprime 265 Smith 265 sphenic 14833 14684570 tribonacci 44 unprimeable 1854 wasteful 44 1854 14833 133496 zygodrome 44