Derangement

In combinatorial mathematics, a derangement is a permutation of the elements of a set in which no element appears in its original position. In other words, a derangement is a permutation that has no fixed points.
The number of derangements of a set of size is known as the subfactorial of or the derangement number or de Montmort number. Notations for subfactorials in common use include,,, or .
For, the subfactorial equals the nearest integer to, where denotes the factorial of and is Euler's number.
The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.

Example

Suppose that a professor gave a test to 4 students – A, B, C, and D – and wants to let them grade each other's tests. How many ways could the professor hand the tests back to the students for grading, such that no student receives their own test back? Out of 24 possible permutations for handing back the tests,
there are only 9 derangements. In every other permutation of this 4-member set, at least one student gets their 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

Counting derangements of a set amounts to the hat-check problem, in which one considers the number of ways in which n hats can be returned to n people such that no hat makes it back to its owner.
Each person may receive any of the n − 1 hats that is not their own. Call the hat which the person P₁ receives h_i and consider h_is owner: P_i receives either P₁'s hat, h₁, or some other. Accordingly, the problem splits into two possible cases:

P_i receives a hat other than h₁. This case is equivalent to solving the problem with n − 1 people and n − 1 hats because for each of the n − 1 people besides P₁ there is exactly one hat from among the remaining n − 1 hats that they may not receive. Another way to see this is to rename h₁ to h_i, where the derangement is more explicit: for any j from 2 to n, P_j cannot receive h_j.
P_i receives h₁. In this case the problem reduces to n − 2 people and n − 2 hats, because P₁ received h_is hat and P_i received h₁'s hat, effectively putting both out of further consideration.

For each of the n − 1 hats that P₁ may receive, the number of ways that P₂, ..., P_n may all receive hats is the sum of the counts for the two cases.
This gives us the solution to the hat-check problem: Stated algebraically, the number !n of derangements of an n-element set is
for,
where and
The number of derangements of small lengths is given in the table below.

n	0	1	2	3	4	5	6	7	8	9	10	11	12	13
!n	1	0	1	2	9	44	265	1,854	14,833	133,496	1,334,961	14,684,570	176,214,841	2,290,792,932

There are various other expressions for, equivalent to the formula given above. These include
for
and
where is the nearest integer function and is the floor function.
Other related formulas include
and
The following recurrence also holds:

Derivation by inclusion–exclusion principle

One may derive a non-recursive formula for the number of derangements of an n-set, as well. For we define to be the set of permutations of objects that fix the object. Any intersection of a collection of of these sets fixes a particular set of objects and therefore contains permutations. There are such collections, so the inclusion–exclusion principle yields
and since a derangement is a permutation that leaves none of the n objects fixed, this implies
On the other hand, since we can choose elements to be in their own place and
derange the other elements in just ways, by definition.

Growth of number of derangements as ''n'' approaches ∞

From
and
by substituting one immediately obtains that
This is the limit of the probability that a randomly selected permutation of a large number of objects is a derangement. The probability converges to this limit extremely quickly as increases, which is why is the nearest integer to 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
statistics of random permutations.

Asymptotic expansion in terms of Bell numbers

An asymptotic expansion for the number of derangements in terms of Bell numbers is as follows:
where is any fixed positive integer, and denotes the -th Bell number. Moreover, the constant implied by the big O-term does not exceed.

Generalizations

The 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 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 surjections A → S, we often wish to know the number of pairs of functions such that f is in U and g is in V, and for all a in A, f ≠ g; in other words, where for each f and g, there exists a derangement φ of S such that f = φ.
Another generalization is the following problem:
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 to 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 n₁ letters X₁, n₂ letters X₂,..., n_r letters X_r, it turns out that the answer has the form
for a certain sequence of polynomials P_n, where P_n has degree n. But the above answer for the case r = 2 gives an orthogonality relation, whence the P_n's are the Laguerre polynomials.
In particular, for the classical derangements, one has that
where is the upper incomplete gamma function.

Computational complexity

It is NP-complete to determine whether a given permutation group contains any derangements.
=1×10⁰1
=1×10⁰ = 111
=1×10⁰0 = 022
=2×10⁰1
=1×10⁰ = 0.536
=6×10⁰2
=2×10⁰≈0.33333 33333424
=2.4×10¹9
=9×10⁰ = 0.3755120
=1.20×10²44
=4.4×10¹≈0.36666 666676720
=7.20×10²265
=2.65×10²≈0.36805 5555675,040
=5.04×10³1,854
≈1.85×10³≈0.36785,71429840,320
≈4.03×10⁴14,833
≈1.48×10⁴≈0.36788 194449362,880
≈3.63×10⁵133,496
≈1.33×10⁵≈0.36787 91887103,628,800
≈3.63×10⁶1,334,961
≈1.33×10⁶≈0.36787 946431139,916,800
≈3.99×10⁷14,684,570
≈1.47×10⁷≈0.36787 9439212479,001,600
≈4.79×10⁸176,214,841
≈1.76×10⁸≈0.36787 94413136,227,020,800
≈6.23×10⁹2,290,792,932
≈2.29×10⁹≈0.36787 944121487,178,291,200
≈8.72×10¹⁰32,071,101,049
≈3.21×10¹⁰≈0.36787 94412151,307,674,368,000
≈1.31×10¹²481,066,515,734
≈4.81×10¹¹≈0.36787 944121620,922,789,888,000
≈2.09×10¹³7,697,064,251,745
≈7.70×10¹²≈0.36787 9441217355,687,428,096,000
≈3.56×10¹⁴130,850,092,279,664
≈1.31×10¹⁴≈0.36787 94412186,402,373,705,728,000
≈6.40×10¹⁵2,355,301,661,033,953
≈2.36×10¹⁵≈0.36787 9441219121,645,100,408,832,000
≈1.22×10¹⁷44,750,731,559,645,106
≈4.48×10¹⁶≈0.36787 94412202,432,902,008,176,640,000
≈2.43×10¹⁸895,014,631,192,902,121
≈8.95×10¹⁷≈0.36787 944122151,090,942,171,709,440,000
≈5.11×10¹⁹18,795,307,255,050,944,540
≈1.88×10¹⁹≈0.36787 94412221,124,000,727,777,607,680,000
≈1.12×10²¹413,496,759,611,120,779,881
≈4.13×10²⁰≈0.36787 944122325,852,016,738,884,976,640,000
≈2.59×10²²9,510,425,471,055,777,937,262
≈9.51×10²¹≈0.36787 9441224620,448,401,733,239,439,360,000
≈6.20×10²³228,250,211,305,338,670,494,289
≈2.28×10²³≈0.36787 944122515,511,210,043,330,985,984,000,000
≈1.55×10²⁵5,706,255,282,633,466,762,357,224
≈5.71×10²⁴≈0.36787 9441226403,291,461,126,605,635,584,000,000
≈4.03×10²⁶148,362,637,348,470,135,821,287,825
≈1.48×10²⁶≈0.36787 944122710,888,869,450,418,352,160,768,000,000
≈1.09×10²⁸4,005,791,208,408,693,667,174,771,274
≈4.01×10²⁷≈0.36787 9441228304,888,344,611,713,860,501,504,000,000
≈3.05×10²⁹112,162,153,835,443,422,680,893,595,673
≈1.12×10²⁹≈0.36787 94412298,841,761,993,739,701,954,543,616,000,000
≈8.84×10³⁰3,252,702,461,227,859,257,745,914,274,516
≈3.25×10³⁰≈0.36787 9441230265,252,859,812,191,058,636,308,480,000,000
≈2.65×10³²97,581,073,836,835,777,732,377,428,235,481
≈9.76×10³¹≈0.36787 94412