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Template:Arrangements

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The {{arrangements}} mathematical function template returns the number of arrangements of 
n, 0   ≤   n   ≤   12,
otherwise returns an error message.

Usage

{{arrangements|a nonnegative integer}}

or (by analogy with subfactorial, i.e. number of derangements: unfortunately the term superfactorial[1] has another meaning)

{{superfactorial|a nonnegative integer}}

or (by analogy with subfactorial, i.e. number of derangements: unfortunately the term superfactorial[1] has another meaning)

{{superfact|a nonnegative integer}}

Valid arguments

Returns the number of arrangements for a nonnegative integer
n, 0   ≤   n   ≤   12,
otherwise returns an error message.

Examples

A000522 The number of arrangements
an
of 
n, n   ≥   0.
{1, 2, 5, 16, 65, 326, 1957, 13700, 109601, 986410, 9864101, 108505112, 1302061345, 16926797486, 236975164805, 3554627472076, 56874039553217, 966858672404690, ...}

Examples with valid argument 
n, 0   ≤   n   ≤   12,
(returns the number of arrangements)

Code Result
{{arrangements|0}} 1
{{arrangements|1}} 2
{{arrangements|2}} 5
{{arrangements|3}} 16
{{arrangements|4}} 65
{{arrangements|5}} 326
{{arrangements|6}} 1957
{{arrangements|7}} 13700
{{arrangements|8}} 109601
{{arrangements|9}} 986410
{{arrangements|10}} 9864101
{{arrangements|11}} 108505112
{{arrangements|12}} 1302061345
The geometric mean of 
an
and 
dn
rounded to nearest integer yields 
n!
, for 
n   ≥   2
, where 
an
and 
dn
are the number of arrangements and the number of derangements of 
n
, 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 
n   ≥   13
(returns a not so user friendly error message)

Code Result
{{arrangements|13}} 16926797486
{{arrangements|14}} 236975164805
{{arrangements|15}} 3554627472076
{{arrangements|16}} 56874039553217
{{arrangements|80}} 7.1526377254407E+35

Examples with invalid argument (returns a user friendly error message)

Code Result
{{arrangements|-1}} Arrangements error: Argument must be a nonnegative integer
{{arrangements|0.5}} Arrangements error: Argument must be a nonnegative integer
{{arrangements|text}} Arrangements error: Argument must be a nonnegative integer
{{arrangements|6 blobs}} Arrangements error: Argument must be a nonnegative integer

Code


{{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}}
}}

Gotcha: using round instead of ceil

Using round instead of ceil gives 3 instead of 2 for 
a1
, otherwise results seem to agree.

{{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}}
}}

Code for {{derangements}}


{{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}}
}}

Notes

  1. 1.0 1.1 Weisstein, Eric W., Superfactorial, from MathWorld—A Wolfram Web Resource.

See also