Orthocentric system

2013-09-04T18:57:38Z

173.18.232.103:

{{For|the psychological condition|psychosis}}
[[File:n! v !n.svg|thumb|300px|Number of possible permutations and derangements of n elements. n! (n factorial) is the number of n-permutations; !n (n subfactorial) is the number of derangements — n-permutations where all of the n elements change their initial places.
{|class="wikitable collapsible collapsed" style="margin:0;" width="100%"
!colspan="4"| Table of values
|-
! <math>n</math>
!nowrap| Permutations, <math>n!</math>
!nowrap| Derangements, <math>!n</math>
! <math>\frac{!n}{n!}</math>
|-
|align="center"| 0
| 1
<span style="font-size:80%; float:right;">=1×10<sup>0</sup></div>
| 1
<span style="font-size:80%; float:right;">=1×10<sup>0</sup></div>
|  = 1
|-
|align="center"| 1
| 1
<span style="font-size:80%; float:right;">=1×10<sup>0</sup></div>
| 0
|  = 0
|-
|align="center"| 2
| 2
<span style="font-size:80%; float:right;">=2×10<sup>0</sup></div>
| 1
<span style="font-size:80%; float:right;">=1×10<sup>0</sup></div>
|  = 0.5
|-
|align="center"| 3
| 6
<span style="font-size:80%; float:right;">=6×10<sup>0</sup></div>
| 2
<span style="font-size:80%; float:right;">=2×10<sup>0</sup></div>
|align="right"| ≈0.33333 33333
|-
|align="center"| 4
| 24
<span style="font-size:80%; float:right;">=2.4×10<sup>1</sup></div>
| 9
<span style="font-size:80%; float:right;">=9×10<sup>0</sup></div>
|  = 0.375
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 5
| 120
<span style="font-size:80%; float:right;">=1.20×10<sup>2</sup></div>
| 44
<span style="font-size:80%; float:right;">=4.4×10<sup>1</sup></div>
|align="right"| ≈0.36666 66667
|-
|align="center"| 6
| 720
<span style="font-size:80%; float:right;">=7.20×10<sup>2</sup></div>
| 265
<span style="font-size:80%; float:right;">=2.65×10<sup>2</sup></div>
|align="right"| ≈0.36805 55556
|-
|align="center"| 7
| 5 040
<span style="font-size:80%; float:right;">≈5.04×10<sup>3</sup></div>
| 1 854
<span style="font-size:80%; float:right;">≈1.85×10<sup>3</sup></div>
|align="right"| ≈0.36785 71429
|-
|align="center"| 8
| 40 320
<span style="font-size:80%; float:right;">≈4.03×10<sup>4</sup></div>
| 14 833
<span style="font-size:80%; float:right;">≈1.48×10<sup>4</sup></div>
|align="right"| ≈0.36788 19444
|-
|align="center"| 9
| 362 880
<span style="font-size:80%; float:right;">≈3.63×10<sup>5</sup></div>
| 133 496
<span style="font-size:80%; float:right;">≈1.33×10<sup>5</sup></div>
|align="right"| ≈0.36787 91887
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 10
| 3 628 800
<span style="font-size:80%; float:right;">≈3.63×10<sup>6</sup></div></div>
| 1 334 961
<span style="font-size:80%; float:right;">≈1.33×10<sup>6</sup></div></div>
|align="right"| ≈0.36787 94643
|-
|align="center"| 11
| 39 916 800
<span style="font-size:80%; float:right;">≈3.99×10<sup>7</sup></div></div>
| 14 684 570
<span style="font-size:80%; float:right;">≈1.47×10<sup>7</sup></div></div>
|align="right"| ≈0.36787 94392
|-
|align="center"| 12
| 479 001 600
<span style="font-size:80%; float:right;">≈4.79×10<sup>8</sup></div></div>
| 176 214 841
<span style="font-size:80%; float:right;">≈1.76×10<sup>8</sup></div></div>
|align="right"| ≈0.36787 94413
|-
|align="center"| 13
| 6 227 020 800
<span style="font-size:80%; float:right;">≈6.23×10<sup>9</sup></div></div>
| 2 290 792 932
<span style="font-size:80%; float:right;">≈2.29×10<sup>9</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 14
| 87 178 291 200
<span style="font-size:80%; float:right;">≈8.72×10<sup>10</sup></div></div>
| 32 071 101 049
<span style="font-size:80%; float:right;">≈3.21×10<sup>10</sup></div></div>
|align="right"| ≈0.36787 94412
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 15
|style="font-size:80%;"| 1 307 674 368 000
<span style="float:right;">≈1.31×10<sup>12</sup></div></div>
|style="font-size:80%;"| 481 066 515 734
<span style="float:right;">≈4.81×10<sup>11</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 16
|style="font-size:80%;"| 20 922 789 888 000
<span style="float:right;">≈2.09×10<sup>13</sup></div></div>
|style="font-size:80%;"| 7 697 064 251 745
<span style="float:right;">≈7.70×10<sup>12</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 17
|style="font-size:80%;"| 355 687 428 096 000
<span style="float:right;">≈3.56×10<sup>14</sup></div></div>
|style="font-size:80%;"| 130 850 092 279 664
<span style="float:right;">≈1.31×10<sup>14</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 18
|style="font-size:80%;"| 6 402 373 705 728 000
<span style="float:right;">≈6.40×10<sup>15</sup></div></div>
|style="font-size:80%;"| 2 355 301 661 033 953
<span style="float:right;">≈2.36×10<sup>15</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 19
|style="font-size:80%;"| 121 645 100 408 832 000
<span style="float:right;">≈1.22×10<sup>17</sup></div></div>
|style="font-size:80%;"| 44 750 731 559 645 106
<span style="float:right;">≈4.48×10<sup>16</sup></div></div>
|align="right"| ≈0.36787 94412
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 20
|style="font-size:80%;"| 2 432 902 008 176 640 000
<span style="float:right;">≈2.43×10<sup>18</sup></div></div>
|style="font-size:80%;"| 895 014 631 192 902 121
<span style="float:right;">≈8.95×10<sup>17</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 21
|style="font-size:80%;"| 51 090 942 171 709 440 000
<span style="float:right;">≈5.11×10<sup>19</sup></div></div>
|style="font-size:80%;"| 18 795 307 255 050 944 540
<span style="float:right;">≈1.88×10<sup>19</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 22
|style="font-size:80%;"| 1 124 000 727 777 607 680 000
<span style="float:right;">≈1.12×10<sup>21</sup></div></div>
|style="font-size:80%;"| 413 496 759 611 120 779 881
<span style="float:right;">≈4.13×10<sup>20</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 23
|style="font-size:80%;"| 25 852 016 738 884 976 640 000
<span style="float:right;">≈2.59×10<sup>22</sup></div></div>
|style="font-size:80%;"| 9 510 425 471 055 777 937 262
<span style="float:right;">≈9.51×10<sup>21</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 24
|style="font-size:80%;"| 620 448 401 733 239 439 360 000
<span style="float:right;">≈6.20×10<sup>23</sup></div></div>
|style="font-size:80%;"| 228 250 211 305 338 670 494 289
<span style="float:right;">≈2.28×10<sup>23</sup></div></div>
|align="right"| ≈0.36787 94412
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 25
|style="font-size:80%;"| 15 511 210 043 330 985 984 000 000
<span style="float:right;">≈1.55×10<sup>25</sup></div></div>
|style="font-size:80%;"| 5 706 255 282 633 466 762 357 224
<span style="float:right;">≈5.71×10<sup>24</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 26
|style="font-size:80%;"| 403 291 461 126 605 635 584 000 000
<span style="float:right;">≈4.03×10<sup>26</sup></div></div>
|style="font-size:80%;"| 148 362 637 348 470 135 821 287 825
<span style="float:right;">≈1.48×10<sup>26</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 27
|style="font-size:80%;"| 10 888 869 450 418 352 160 768 000 000
<span style="float:right;">≈1.09×10<sup>28</sup></div></div>
|style="font-size:80%;"| 4 005 791 208 408 693 667 174 771 274
<span style="float:right;">≈4.01×10<sup>27</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 28
|style="font-size:80%;"| 304 888 344 611 713 860 501 504 000 000
<span style="float:right;">≈3.05×10<sup>29</sup></div></div>
|style="font-size:80%;"| 112 162 153 835 443 422 680 893 595 673
<span style="float:right;">≈1.12×10<sup>29</sup></div></div>
|align="right"| ≈0.36787 94412
|-
|align="center"| 29
|style="font-size:80%;"| 8 841 761 993 739 701 954 543 616 000 000
<span style="float:right;">≈8.84×10<sup>30</sup></div></div>
|style="font-size:80%;"| 3 252 702 461 227 859 257 745 914 274 516
<span style="float:right;">≈3.25×10<sup>30</sup></div></div>
|align="right"| ≈0.36787 94412
|-style="border-top:2px solid #aaaaaa;"
|align="center"| 30
|style="font-size:80%;"| 265 252 859 812 191 058 636 308 480 000 000
<span style="float:right;">≈2.65×10<sup>32</sup></div></div>
|style="font-size:80%;"| 97 581 073 836 835 777 732 377 428 235 481
<span style="float:right;">≈9.76×10<sup>31</sup></div></div>
|align="right"| ≈0.36787 94412
|}]]

In [[combinatorics|combinatorial]] [[mathematics]], a '''derangement''' is a [[permutation]] of the elements of a [[set (mathematics)|set]] such that none of the elements appear in their original position.

The numbers of derangements !''n'' for sets of size ''n'' are called "de Montmort numbers" or "derangement numbers" (and can be generalized to [[rencontres numbers]]); the '''subfactorial''' function (not to be confused with the [[factorial]] ''n''!) maps ''n'' to !''n''.<ref>The name "subfactorial" originates with [[William Allen Whitworth]]; see {{citation|title=A History of Mathematical Notations: Two Volumes in One|first=Florian|last=Cajori|authorlink=Florian Cajori|publisher=Cosimo, Inc.,|year=2011|isbn=9781616405717|page=77|url=http://books.google.com/books?id=gxrO8ZnMK_YC&pg=RA1-PA77}}.</ref> No standard notation for subfactorials is agreed upon, and ''n''¡ is sometimes used instead of !''n''.<ref>Ronald L. Graham, Donald E. Knuth, Oren Patashnik, ''Concrete Mathematics'' (1994), Addison–Wesley, Reading MA. ISBN 0-201-55802-5</ref>

The problem of counting derangements was first considered by [[Pierre Raymond de Montmort]]<ref>de Montmort, P. R. (1708). ''Essay d'analyse sur les jeux de hazard''. Paris: Jacque Quillau. ''Seconde Edition, Revue & augmentée de plusieurs Lettres''. Paris: Jacque Quillau. 1713.</ref> in 1708; he solved it in 1713, as did [[Nicolaus I Bernoulli|Nicholas Bernoulli]] at about the same time.

== Example ==

Suppose that a professor has had 4 of his students – student A, student B, student C, and student D - take a test and wants to let his students grade each other's tests. Of course, no student should grade his or her own test. How many ways could the professor hand the tests back to the students for grading, such that no student received his or her own test back? Out of [[v:Symmetric_group_S4#tables|24 possible permutations]] (4!) for handing back the tests, there are only 9 derangements:

:BADC, BCDA, BDAC,
:CADB, CDAB, CDBA,
:DABC, DCAB, DCBA.

In every other permutation of this 4-member set, at least one student gets his or her own test back.

Another version of the problem arises when we ask for the number of ways ''n'' letters, each addressed to a different person, can be placed in ''n'' pre-addressed envelopes so that no letter appears in the correctly addressed envelope.

== Counting derangements ==
Suppose that there are ''n'' persons numbered 1, 2, ..., ''n''. Let there be ''n'' hats also numbered 1, 2, ..., ''n''. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that the first person takes hat ''i''. There are ''n'' − 1 ways for the first person to make such a choice. There are now two possibilities, depending on whether or not person ''i'' takes hat 1 in return:
#Person ''i'' does not take the hat 1. This case is equivalent to solving the problem with ''n'' − 1 persons ''n'' − 1 hats: each of the remaining ''n'' − 1 people has precisely 1 forbidden choice from among the remaining ''n'' − 1 hats (''i'''s forbidden choice is hat 1).
#Person ''i'' takes the hat 1. Now the problem reduces to ''n'' − 2 persons and ''n'' − 2 hats.
From this, the following relation is derived:

:<math>!n = (n - 1) (!(n-1) + !(n-2)).\,</math>

with the starting values !0 = 1 and !1 = 0.

Notice that this same recurrence formula also works for factorials with different starting values. That is 0! = 1, 1! = 1 and

:<math>n! = (n - 1) ((n-1)! + (n-2)!)\,</math>

which is helpful in proving the limit relationship with ''e'' below.

Also, the following formulas are known:<ref>Hassani, M. "Derangements and Applications." J. Integer Seq. 6, No. 03.1.2, 1–8, 2003</ref>

:<math>!n = n! \sum_{i=0}^n \frac{(-1)^i}{i!},</math>

:<math>!n = \left\lfloor\frac{n!}{e}+\frac{1}{2}\right\rfloor , \quad n\geq 1,</math>

:<math>!n = \left[ \frac{n!}{e} \right] , \quad n\geq 1</math>
Where '''[x]''' is the nearest integer function.

:<math>!n = \left\lfloor(e+e^{-1})n!\right\rfloor-\lfloor en!\rfloor , \quad n\geq 2,</math>

:<math>!n = n! - \sum_{i=1}^n {n \choose i} \cdot !(n-i),</math>

where <math>\left\lfloor x \right\rfloor</math> is the [[floor function]]. The following recurrence relationship also holds:<ref>See the notes for {{OEIS|id=A000166}}.</ref>

:<math>!n = n[!(n-1)] + (-1)^n</math>

Starting with ''n'' = 0, the numbers of derangements of ''n'' are:

:1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, ... {{OEIS|id=A000166}}.

These numbers are also called '''subfactorial''' or '''[[rencontres numbers]]'''.

Perhaps a more well-known method of counting derangements uses the [[inclusion-exclusion principle#Examples|inclusion-exclusion principle]].

==Limit of ratio of derangement to permutation as ''n'' approaches ∞==

Using this recurrence, it can be shown that, in the limit,

:<math>\lim_{n\to\infty} {!n \over n!} = {1 \over e} \approx 0.3679\dots.</math>

This is the limit of the [[probability]] ''p''<sub>''n''</sub> = ''d''<sub>''n''</sub>/''n''! that a randomly selected permutation is a derangement. The probability converges to this limit quickly as ''n'' increases. The above [[semi-log]] graph shows that the derangement graph lags the permutation graph by an almost constant value.

More information about this calculation and the above limit may be found in the article on the
[[Random permutation statistics#Number of permutations that are derangements|statistics of random permutations]].

== Generalizations ==

The [[rencontres numbers|problème des rencontres]] asks how many permutations of a size-''n'' set have exactly ''k'' fixed points.

Derangements are an example of the wider field of constrained permutations. For example, the ''[[ménage problem]]'' asks if ''n'' opposite-sex couples are seated man-woman-man-woman-... around a circular table, how many ways can they be seated so that nobody is seated next to his or her partner?

More formally, given sets ''A'' and ''S'', and some sets ''U'' and ''V'' of [[surjection]]s ''A'' → ''S'', we often wish to know the number of pairs of functions (''f'', ''g'') such that ''f'' is in ''U'' and ''g'' is in ''V'', and for all ''a'' in ''A'', ''f''(''a'') ≠ ''g''(''a''); in other words, where for each ''f'' and ''g'', there exists a derangement φ of ''S'' such that ''f''(''a'') = φ(''g''(''a'')).

Another generalization is the following problem:

:''How many anagrams with no fixed letters of a given word are there?''

For instance, for a word made of only two different letters, say ''n'' letters A and ''m'' letters B, the answer is, of course, 1 or 0 according whether ''n'' = ''m'' or not, for the only way to form an anagram without fixed letters is to exchange all the ''A'' with ''B'', which is possible if and only if ''n'' = ''m''. In the general case, for a word with ''n''<sub>1</sub> letters ''X''<sub>1</sub>, ''n''<sub>2</sub> letters ''X''<sub>2</sub>, ..., ''n''<sub>''r''</sub> letters ''X''<sub>''r''</sub> it turns out (after a proper use of the [[inclusion-exclusion]] formula) that the answer has the form:

:<math>\int_0^\infty P_{n_1} (x) P_{n_2}(x)\cdots P_{n_r}(x) e^{-x}\, dx,</math>

for a certain sequence of polynomials ''P''<sub>''n''</sub>, where ''P''<sub>''n''</sub> has degree ''n''. But the above answer for the case ''r'' = 2 gives an orthogonality relation, whence the ''P''<sub>''n''</sub><nowiki>'</nowiki>s are the [[Laguerre polynomials]] ([[up to]] a sign that is easily decided).<ref>{{cite journal|last=Even|first=S.|coauthors=J. Gillis|title=Derangements and Laguerre polynomials|journal=Mathematical Proceedings of the Cambridge Philosophical Society|year=1976|volume=79|issue=01|pages=135–143|doi=10.1017/S0305004100052154|url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2128316|accessdate=27 December 2011}}</ref>

==Computational complexity==
It is [[NP-complete]] to determine whether a given [[permutation group]] (described by a given set of permutations that generate it) contains any derangements.<ref>{{citation
| last = Lubiw | first = Anna | authorlink = Anna Lubiw
| doi = 10.1137/0210002
| issue = 1
| journal = [[SIAM Journal on Computing]]
| mr = 605600
| pages = 11–21
| title = Some NP-complete problems similar to graph isomorphism
| volume = 10
| year = 1981}}. {{citation
| last = Babai | first = László | authorlink = László Babai
| contribution = Automorphism groups, isomorphism, reconstruction
| location = Amsterdam
| mr = 1373683
| pages = 1447–1540
| publisher = Elsevier
| quotation = A surprising result of Anna Lubiw asserts that the following problem is NP-complete: Does a given permutation group have a fixed-point-free element?
| title = Handbook of combinatorics, Vol. 1, 2
| url = http://people.cs.uchicago.edu/~laci/handbook/handbookchapter27.pdf
| year = 1995}}.</ref>

== References ==

{{Reflist}}

== External links ==
* {{cite web
| author = Baez, John
| title = Let's get deranged!
| url = http://math.ucr.edu/home/baez/qg-winter2004/derangement.pdf
| year = 2003
| authorlink = John Baez}}
* {{cite web
| year = 1985
| url = http://www.math.dartmouth.edu/~doyle/docs/menage/menage/menage.html
| title = Non-sexist solution of the ménage problem
| author = Bogart, Kenneth P. and Doyle, Peter G.}}
* {{cite web
| author = Dickau, Robert M.
| title = Derangement diagrams
| url = http://mathforum.org/advanced/robertd/derangements.html
| work = [http://mathforum.org/advanced/robertd/index.htmlMathematical Figures Using ''Mathematica'']}}
* {{cite web
| author = Hassani, Mehdi
| title = Derangements and Applications
| publisher = Journal of Integer Sequences (JIS), Volume 6, Issue 1, Article 03.1.2, 2003
| url = http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Hassani/hassani5.html}}
* {{cite web
| author = Weisstein, Eric W
| authorlink = Eric W. Weisstein
| title = Derangement
| publisher = MathWorld–A Wolfram Web Resource
| url = http://mathworld.wolfram.com/Derangement.html}}
* {{cite web
| author = Debra K. Borkovitz
| title = Derangements and the Inclusion-Exclusion Principle
| publisher = Articles, Associate Professor of Mathematics, [[Wheelock College]]
| url = http://faculty.wheelock.edu/dborkovitz/articles/ngm6.htm}}

[[Category:Permutations]]
[[Category:Fixed points (mathematics)]]
[[Category:Integer sequences]]

[[es:Subfactorial]]
[[fr:Analogues de la factorielle#Sous-factorielle]]

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Orthocentric system