The
{{derangements}} mathematical function template returns the
subfactorial , i.e. the
number of derangements of
otherwise returns an error message.
Usage
- {{derangements|a nonnegative integer}}
or
- {{subfactorial|a nonnegative integer}}
or
- {{subfact|a nonnegative integer}}
or
- {{!n|a nonnegative integer}}
Valid argument
Returns
for a
nonnegative integer otherwise returns an error message.
Examples
A000166 Subfactorial numbers, or
number of derangements: number of permutations of
elements with no rencontres, i.e. no fixed points.
- {1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, ...}
Examples with valid argument (returns the number of derangements)
Code
|
Result
|
{{derangements|12}} |
176214841
|
{{subfactorial|12}} |
176214841
|
{{subfact|12}} |
176214841
|
{{!n|12}} |
176214841
|
Code
|
Result
|
{{!n|0}} |
1
|
{{!n|1}} |
0
|
{{!n|2}} |
1
|
{{!n|3}} |
2
|
{{!n|4}} |
9
|
{{!n|5}} |
44
|
{{!n|6}} |
265
|
{{!n|7}} |
1854
|
{{!n|8}} |
14833
|
{{!n|9}} |
133496
|
{{!n|10}} |
1334961
|
{{!n|11}} |
14684570
|
{{!n|12}} |
176214841
|
The
geometric mean of
and
rounded to nearest integer yields
, for
, where
and
are the
number of arrangements and the
number of derangements of
, respectively.
Code
|
Result
|
|
Code
|
Result
|
{{root| {{arrangements|0}} * {{derangements|0}} }}
|
2√ 1 * 1
|
|
{{n!|0}}
|
1
|
{{root| {{arrangements|1}} * {{derangements|1}} }}
|
2√ 2 * 0
|
|
{{n!|1}}
|
1
|
{{root| {{arrangements|2}} * {{derangements|2}} }}
|
2√ 5 * 1
|
|
{{n!|2}}
|
2
|
{{root| {{arrangements|3}} * {{derangements|3}} }}
|
2√ 16 * 2
|
|
{{n!|3}}
|
6
|
{{root| {{arrangements|4}} * {{derangements|4}} }}
|
2√ 65 * 9
|
|
{{n!|4}}
|
24
|
{{root| {{arrangements|5}} * {{derangements|5}} }}
|
2√ 326 * 44
|
|
{{n!|5}}
|
120
|
{{root| {{arrangements|6}} * {{derangements|6}} }}
|
2√ 1957 * 265
|
|
{{n!|6}}
|
720
|
{{root| {{arrangements|7}} * {{derangements|7}} }}
|
2√ 13700 * 1854
|
|
{{n!|7}}
|
5040
|
{{root| {{arrangements|8}} * {{derangements|8}} }}
|
2√ 109601 * 14833
|
|
{{n!|8}}
|
40320
|
{{root| {{arrangements|9}} * {{derangements|9}} }}
|
2√ 986410 * 133496
|
|
{{n!|9}}
|
362880
|
Examples with valid, but out of range, argument (returns a not so user friendly error message)
Code
|
Result
|
{{!n|13}} |
2290792932
|
{{!n|14}} |
32071101049
|
{{!n|15}} |
481066515734
|
{{!n|16}} |
7697064251745
|
{{!n|80}} |
9.680042524614E+34
|
Examples with invalid argument (returns a user friendly error message)
Code
|
Result
|
{{!n|-1}} |
Derangements error: Argument must be a nonnegative integer
|
{{!n|0.5}} |
Derangements error: Argument must be a nonnegative integer
|
{{!n|text}} |
Derangements error: Argument must be a nonnegative integer
|
{{!n|6 blobs}} |
Derangements error: Argument must be a nonnegative integer
|
Code
{{ifint| ( {{{1|empty}}} )
| {{#ifexpr: ( {{{1}}} ) >= 0
| {{#ifexpr: ( {{{1}}} ) = 0
| 1
| {{expr| ( {{~Pochhammer| 1 | ( {{{1}}} ) | 1 }} / e ) round 0 }}
}}
| {{error| Derangements error: Argument must be a nonnegative integer}}
}}
| {{error| Derangements error: Argument must be a nonnegative integer}}
}}
{{ifint| ( {{{1|empty}}} )
| {{#ifexpr: ( {{{1}}} ) >= 0
| {{#ifexpr: ( {{{1}}} ) = 0
| 1
| {{expr| floor ( {{~Pochhammer| 1 | ( {{{1}}} ) | 1 }} * e ) }}
}}
| {{error| Arrangements error: Argument must be a nonnegative integer}}
}}
| {{error| Arrangements error: Argument must be a nonnegative integer}}
}}
See also