Table of values | |||
---|---|---|---|
Permutations, | Derangements, | ||
0 | 1 =1×10 | 1 =1×10 | = 1 |
1 | 1 =1×10 | 0 | = 0 |
2 | 2 =2×10 | 1 =1×10 | = 0.5 |
3 | 6 =6×10 | 2 =2×10 | ≈0.33333 33333 |
4 | 24 =2.4×10 | 9 =9×10 | = 0.375 |
5 | 120 =1.20×10 | 44 =4.4×10 | ≈0.36666 66667 |
6 | 720 =7.20×10 | 265 =2.65×10 | ≈0.36805 55556 |
7 | 5,040 =5.04×10 | 1,854 ≈1.85×10 | ≈0.36785,71429 |
8 | 40,320 ≈4.03×10 | 14,833 ≈1.48×10 | ≈0.36788 19444 |
9 | 362,880 ≈3.63×10 | 133,496 ≈1.33×10 | ≈0.36787 91887 |
10 | 3,628,800 ≈3.63×10 | 1,334,961 ≈1.33×10 | ≈0.36787 94643 |
11 | 39,916,800 ≈3.99×10 | 14,684,570 ≈1.47×10 | ≈0.36787 94392 |
12 | 479,001,600 ≈4.79×10 | 176,214,841 ≈1.76×10 | ≈0.36787 94413 |
13 | 6,227,020,800 ≈6.23×10 | 2,290,792,932 ≈2.29×10 | ≈0.36787 94412 |
14 | 87,178,291,200 ≈8.72×10 | 32,071,101,049 ≈3.21×10 | ≈0.36787 94412 |
15 | 1,307,674,368,000 ≈1.31×10 | 481,066,515,734 ≈4.81×10 | ≈0.36787 94412 |
16 | 20,922,789,888,000 ≈2.09×10 | 7,697,064,251,745 ≈7.70×10 | ≈0.36787 94412 |
17 | 355,687,428,096,000 ≈3.56×10 | 130,850,092,279,664 ≈1.31×10 | ≈0.36787 94412 |
18 | 6,402,373,705,728,000 ≈6.40×10 | 2,355,301,661,033,953 ≈2.36×10 | ≈0.36787 94412 |
19 | 121,645,100,408,832,000 ≈1.22×10 | 44,750,731,559,645,106 ≈4.48×10 | ≈0.36787 94412 |
20 | 2,432,902,008,176,640,000 ≈2.43×10 | 895,014,631,192,902,121 ≈8.95×10 | ≈0.36787 94412 |
21 | 51,090,942,171,709,440,000 ≈5.11×10 | 18,795,307,255,050,944,540 ≈1.88×10 | ≈0.36787 94412 |
22 | 1,124,000,727,777,607,680,000 ≈1.12×10 | 413,496,759,611,120,779,881 ≈4.13×10 | ≈0.36787 94412 |
23 | 25,852,016,738,884,976,640,000 ≈2.59×10 | 9,510,425,471,055,777,937,262 ≈9.51×10 | ≈0.36787 94412 |
24 | 620,448,401,733,239,439,360,000 ≈6.20×10 | 228,250,211,305,338,670,494,289 ≈2.28×10 | ≈0.36787 94412 |
25 | 15,511,210,043,330,985,984,000,000 ≈1.55×10 | 5,706,255,282,633,466,762,357,224 ≈5.71×10 | ≈0.36787 94412 |
26 | 403,291,461,126,605,635,584,000,000 ≈4.03×10 | 148,362,637,348,470,135,821,287,825 ≈1.48×10 | ≈0.36787 94412 |
27 | 10,888,869,450,418,352,160,768,000,000 ≈1.09×10 | 4,005,791,208,408,693,667,174,771,274 ≈4.01×10 | ≈0.36787 94412 |
28 | 304,888,344,611,713,860,501,504,000,000 ≈3.05×10 | 112,162,153,835,443,422,680,893,595,673 ≈1.12×10 | ≈0.36787 94412 |
29 | 8,841,761,993,739,701,954,543,616,000,000 ≈8.84×10 | 3,252,702,461,227,859,257,745,914,274,516 ≈3.25×10 | ≈0.36787 94412 |
30 | 265,252,859,812,191,058,636,308,480,000,000 ≈2.65×10 | 97,581,073,836,835,777,732,377,428,235,481 ≈9.76×10 | ≈0.36787 94412 |
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 n is known as the subfactorial of n or the n-th derangement number or n-th de Montmort number (after Pierre Remond de Montmort). Notations for subfactorials in common use include !n,Dn, dn, or n¡. [1] [2]
For n > 0, the subfactorial !n equals the nearest integer to n!/e, where n! denotes the factorial of n and e is Euler's number. [3]
The problem of counting derangements was first considered by Pierre Raymond de Montmort in his Essay d'analyse sur les jeux de hazard. [4] in 1708; he solved it in 1713, as did Nicholas Bernoulli at about the same time.
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. Of course, no student should grade their own test. How many ways could the professor hand the tests back to the students for grading, such that no student received their own test back? Out of 24 possible permutations (4!) for handing back the tests,
ABCD, | ABDC, | ACBD, | ACDB, | ADBC, | ADCB, |
BACD, | BADC, | BCAD, | BCDA, | BDAC, | BDCA, |
CABD, | CADB, | CBAD, | CBDA, | CDAB, | CDBA, |
DABC, | DACB, | DBAC, | DBCA, | DCAB, | DCBA. |
there are only 9 derangements (shown in blue italics above). In every other permutation of this 4-member set, at least one student gets their own test back (shown in bold red).
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 of a set amounts to the hat-check problem, in which one considers the number of ways in which n hats (call them h1 through hn) can be returned to n people (P1 through Pn) such that no hat makes it back to its owner. [5]
Each person may receive any of the n − 1 hats that is not their own. Call the hat which the person P1 receives hi and consider hi's owner: Pi receives either P1's hat, h1, or some other. Accordingly, the problem splits into two possible cases:
For each of the n − 1 hats that P1 may receive, the number of ways that P2, ..., Pn 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
where and . [6]
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 !n, equivalent to the formula given above. These include
and
where is the nearest integer function and is the floor function. [3] [6]
Other related formulas include [3] [7]
and
The following recurrence also holds: [6]
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 n objects that fix the -th object. Any intersection of a collection of i of these sets fixes a particular set of i 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 n - i elements to be in their own place and derange the other i elements in just !i ways, by definition. [8]
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 n increases, which is why !n is the nearest integer to n!/e. 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.
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 . [9]
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 (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:
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, ..., nr letters Xr, it turns out (after a proper use of the inclusion-exclusion formula) that the answer has the form
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 Pn's are the Laguerre polynomials (up to a sign that is easily decided). [10]
In particular, for the classical derangements, one has that
where is the upper incomplete gamma function.
It is NP-complete to determine whether a given permutation group (described by a given set of permutations that generate it) contains any derangements. [11]
In mathematics, the floor function (or greatest integer function) is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).
Golomb coding is a lossless data compression method using a family of data compression codes invented by Solomon W. Golomb in the 1960s. Alphabets following a geometric distribution will have a Golomb code as an optimal prefix code, making Golomb coding highly suitable for situations in which the occurrence of small values in the input stream is significantly more likely than large values.
In computer science, the Akra–Bazzi method, or Akra–Bazzi theorem, is used to analyze the asymptotic behavior of the mathematical recurrences that appear in the analysis of divide and conquer algorithms where the sub-problems have substantially different sizes. It is a generalization of the master theorem for divide-and-conquer recurrences, which assumes that the sub-problems have equal size. It is named after mathematicians Mohamad Akra and Louay Bazzi.
In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as
In number theory, the Mertens function is defined for all positive integers n as
In number theory, the integer square root (isqrt) of a non-negative integer n is the non-negative integer m which is the greatest integer less than or equal to the square root of n,
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
In graph theory, an m-ary tree is an arborescence in which each node has no more than m children. A binary tree is the special case where m = 2, and a ternary tree is another case with m = 3 that limits its children to three.
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.
The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published.
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements. There are several proofs of the theorem.
A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set { 1, ..., n } with specified numbers of fixed points: in other words, partial derangements. For n ≥ 0 and 0 ≤ k ≤ n, the rencontres number Dn, k is the number of permutations of { 1, ..., n } that have exactly k fixed points.
The statistics of random permutations, such as the cycle structure of a random permutation are of fundamental importance in the analysis of algorithms, especially of sorting algorithms, which operate on random permutations. Suppose, for example, that we are using quickselect to select a random element of a random permutation. Quickselect will perform a partial sort on the array, as it partitions the array according to the pivot. Hence a permutation will be less disordered after quickselect has been performed. The amount of disorder that remains may be analysed with generating functions. These generating functions depend in a fundamental way on the generating functions of random permutation statistics. Hence it is of vital importance to compute these generating functions.
In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.
In mathematics, Abel's summation formula, introduced by Niels Henrik Abel, is intensively used in analytic number theory and the study of special functions to compute series.
In probability theory and statistics, the Conway–Maxwell–Poisson distribution is a discrete probability distribution named after Richard W. Conway, William L. Maxwell, and Siméon Denis Poisson that generalizes the Poisson distribution by adding a parameter to model overdispersion and underdispersion. It is a member of the exponential family, has the Poisson distribution and geometric distribution as special cases and the Bernoulli distribution as a limiting case.
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.
In analytic number theory, a Dirichlet series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little known, or at least often forgotten about, way of expressing formulas for arithmetic functions and their summatory functions is to perform an integral transform that inverts the operation of forming the DGF of a sequence. This inversion is analogous to performing an inverse Z-transform to the generating function of a sequence to express formulas for the series coefficients of a given ordinary generating function.
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?.