# Dictionary Definition

derangement

### Noun

1 a state of mental disturbance and
disorientation [syn: mental
unsoundness, unbalance]

2 the act of disturbing the mind or body; "his
carelessness could have caused an ecological upset"; "she was
unprepared for this sudden overthrow of their normal way of living"
[syn: upset, overthrow]

# User Contributed Dictionary

## English

### Noun

- The property of being deranged

# Extensive Definition

In combinatorial mathematics, a derangement
is a permutation in
which none of the elements of the set appear in their original
positions. That is, it is a bijection φ from a set S into itself with no fixed
points: for all x in S, φ(x) ≠ x. A frequent problem is
to count the number of derangements as a function of the number of elements of the
set, often with additional constraints; these numbers are called
subfactorials and
are a special case of the rencontres
numbers. The problem of counting derangements was first
considered by
Pierre Raymond de Montmort in 1708; he solved it in 1713, as
did Nicholas
Bernoulli at about the same time.

## Example

Suppose that a professor has graded 4 tests for 4
students. The first student, student A, received an "A" on the
test, student B received a "B", and so on. However, the professor
mixed up the tests when handing them back, and now none of the
students has the correct test. How many ways could the professor
have mixed them all up in this way? Out of 24 possible permutations
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 the right test.

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

One approach to counting the derangements of n
elements is to use induction.
First, note that if φn is any derangement of the natural
numbers , then for some k in , φn(n) = k. Then if we
let (k, n) be the permutation of which swaps k and n, and
we let φn − 1 be the composition
((k, n) o φn); then φn−1(n) = n,
and either:

- φn − 1(k) ≠ k, so φn − 1 is a derangement of , or
- φn−1(k) = k, and for all x ≠ k, φn−1(x) ≠ x.

As examples of these two cases, consider the
following two derangements of 6 elements as we perform the above
described swaps:

- 514623 → (51432)6; and
- 315624 → (31542)6 → (3142)56

The above described correspondences are 1-to-1.
The converse is also true; there are exactly
(n − 1) ways of converting any
derangement of n − 1 elements into a
derangement of n elements, and (n − 1)
ways of converting any derangement of
n − 2 elements into a derangement of n
elements. For example, if n = 6 and k = 4, we can perform the
following conversions of derangements of length 5 and 4,
respectively

- 51432 → 514326 → 514623; and
- 3142 → 31542 → 315426 → 315624

Thus, if we write dn as the number of
derangements of n letters, and we define d0 = 1, d1 = 0; then dn
satisfies the recurrence:

- d_n = (n - 1) (d_ + d_)\,

and also

- d_n = n d_ + (-1)^ , \quad n\geq 1

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

- n! = (n - 1) ((n-1)! + (n-2)!)\,

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

Also, the following formulas are known :

- d_n = n! \sum_^n \frac

- d_n = \left\lfloor\frac+\frac\right\rfloor = \left(\text\frac\right), \quad n\geq 1

- d_n = \left\lfloor(e+e^)n!\right\rfloor-\lfloor en!\rfloor , \quad n\geq 2.

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

- 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, ... .

These numbers are also called subfactorial or rencontres
numbers.

## Limit as n approaches ∞

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

- \lim_ = \approx 0.3679\dots.

This is the limit of the probability pn = dn/n! that
a randomly selected permutation is a derangement. The probability
approaches this limit quite quickly.

Perhaps a more well-known method of counting
derangements uses the
inclusion-exclusion principle.

More information about this calculation and the
above limit may be found on the page on the
statistics of random permutations.

## 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 married couples are seated
boy-girl-boy-girl-... around a circular table, how many ways can
they be seated so that no man is seated next to his wife?

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
(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 n1 letters X1, n2 letters X2, ..., nr letters
Xr it turns out (after a proper use of the inclusion-exclusion
formula) that the answer has the form:

- \int_0^\infty P_ (x) P_(x)\cdots P_(x) e^\, dx,

for a certain sequence of polynomials Pn, where
Pn has degree n. But the above answer for the case r = 2 gives an
orthogonality relation, whence the Pns are the Laguerre
polynomials (up to a sign that
is easily decided).

## References

- 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.

## External links

derangement in German: Derangement

derangement in Italian: Dismutazione
(matematica)

derangement in Japanese: 完全順列

derangement in Russian: Беспорядок
(перестановка)

derangement in Slovak: Dismutácia
(matematika)

derangement in Chinese: 亂序

# Synonyms, Antonyms and Related Words

aberrance, aberrancy, aberration, abnormality, abnormity, alienation, amorphism, anomalism, anomalousness, anomaly, brain damage, brainsickness, clouded
mind, confusion,
convulsion, craziness, daftness, dementedness, dementia, deviation, difference, disarrangement, disarray, disarticulation,
discomfiture,
discomposure,
disconcertedness,
disharmony, dishevelment, disintegration, disjunction, dislocation, disorder, disorderliness, disorganization,
disorientation,
disproportion,
disruption, distraction, disturbance, divergence, eccentricity, entropy, erraticism, folie, furor, haphazardness, heteromorphism, incoherence, indiscriminateness,
inferiority,
inharmonious harmony, insaneness, insanity, irrationality, irregularity, loss of mind,
loss of reason, lunacy,
madness, mania, mental deficiency, mental
derangement, mental disease, mental disorder, mental disturbance,
mental illness, mental instability, mental sickness, mind
overthrown, mindsickness, misarrangement, monstrosity, most admired
disorder, nonsymmetry, nonuniformity, oddness, perturbation, pixilation, possession, promiscuity, promiscuousness,
psychopathy,
queerness, rabidness, randomness, reasonlessness, senselessness, shattered
mind, shuffling, sick
mind, sickness,
strangeness,
subnormality,
superiority,
teratism, turbulence, unbalance, unbalanced mind,
unnaturalism,
unnaturalness,
unsaneness, unsound
mind, unsoundness,
unsoundness of mind, unsymmetry, ununiformity, upset, witlessness