Subfactorial

The subfactorial of the number n (symbol :! N ) is defined as the number of disorders of order n , that is, permutations of order n without fixed points. The name of the subfactorial comes from the analogy with factorial, which determines the total number of permutations.

In particular ,! N is the number of ways to put n letters in n envelopes (one in each), so that no one gets into the corresponding envelope (the so-called Letter Problem).

Explicit Formula

The subfactorial can be calculated using the inclusion-exclusion principle:

Other formulas [edit]

• where denotes an incomplete gamma function ( English ), and e is a mathematical constant;
• where denotes the integer closest to x .
• (according to Mehdi Hassani ) where denotes the integer part of a number.
• Formal identities are true: and where need to understand how , but - as .

Table of Values

! 1 = 0

! 2 = 1

! 3 = 2

! 4 = 9

! 5 = 44

! 6 = 265

! 7 = 1 854

! 8 = 14,833

! 9 = 133 496

! 10 = 1,334,961

! 11 = 14,684,570

! 12 = 176 214 841

! 13 = 2 290 792 932

! 14 = 32 071 101 049

! 15 = 481 066 515 734

! 16 = 7 697 064 251 745

! 17 = 130,850,092,279,664

! 18 = 2 355 301 661 033 953

! 19 = 44 750 731 559 645 106

! 20 = 895 014 631 192 902 121

! 21 = 18,795,307 255,050,944,540

sequence A000166 in OEIS

Properties [edit]

• 
• (factorial has the same property)
• 

1, 1, 3, 11, 53, 309, 2119, ... (sequence A000255 in OEIS)

• The number 148349 is a subfactorion, that is, it is equal to the sum of the subfactorials of its numbers (analogue of the factorion):

(found JS Madachy, 1979)

• Subfactorial is sometimes allowed in math games such as obtaining different results from certain numbers (for example, the game Four Four is known, where equality! 4 = 9 can be useful).