Derangement

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 (call them h₁ through h_n) can be returned to n people (P₁ through P_n) 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_i's owner: P_i receives either P₁'s hat, h₁, or some other. Accordingly, the problem splits into two possible cases:

1. 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 (for any P_j besides P_i, the unreceivable hat is h_j, while for P_i it is h₁). 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.
2. P_i receives h₁. In this case the problem reduces to n − 2 people and n − 2 hats, because P₁ received h_i's 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 D_n of derangements of an n-element set is $${\displaystyle D_n=\left(n-1\right)(D_{n-1}+D_{n-2})\text{for}n\ge 2,}$$ where D₀ = 1 and D₁ = 0.

The number of derangements of small lengths is given in the table below.

There are various other expressions for D_n, equivalent to the formula given above. These include $${\displaystyle D_n=n!\sum _{i=0}^n\frac{(-1{)}^i}{i!}\text{for}n\ge 0}$$ and $${\displaystyle D_n=\left[\frac{n!}{e}\right]=\lfloor \frac{n!}{e}+\frac{1}{2}\rfloor \text{for}n\ge 1,}$$ where [x] is the nearest integer function and ⌊x⌋ is the floor function.

Other related formulas include $${\displaystyle \begin{array}{cccc}{D}_n& =\lfloor \frac{n!+1}{e}\rfloor & \text{for}n& \ge 1,\\ D_n& =\lfloor \left(e+e^{-1}\right)n!\rfloor -\lfloor en!\rfloor & \text{for}n& \ge 2,\\ D_n& =n!-\sum _{i=1}^n(\genfrac{}{}{0ex}{}{n}{i})\cdot D_{n-i}& \text{for}n& \ge 1.\end{array}}$$

The following recurrence also holds: $${\displaystyle D_n=\{\begin{array}{ll}1& \text{if }n=0,\\ n\cdot D_{n-1}+(-1{)}^n& \text{if }n>0.\end{array}}$$

Derivation by inclusion–exclusion principle

One may derive a non-recursive formula for the number of derangements of an n-set, as well. For 1 ≤ k ≤ n we define S_k to be the set of permutations of n objects that fix the k th object. Any intersection of a collection of i of these sets fixes a particular set of i objects and therefore contains (n − i)! permutations. There are $(\genfrac{}{}{0ex}{}{n}{i})$ such collections, so the inclusion–exclusion principle yields $${\displaystyle \begin{array}{cc}\left|S_1\cup \cdots \cup S_n\right|& =\sum _i\left|S_i\right|-\sum _{i<j}\left|S_i\cap S_j\right|+\sum _{i<j<k}\left|S_i\cap S_j\cap S_k\right|+\cdots +(-1{)}^{n+1}\left|S_1\cap \cdots \cap S_n\right|\\ & =(\genfrac{}{}{0ex}{}{n}{1})(n-1)!-(\genfrac{}{}{0ex}{}{n}{2})(n-2)!+(\genfrac{}{}{0ex}{}{n}{3})(n-3)!-\cdots +(-1{)}^{n+1}(\genfrac{}{}{0ex}{}{n}{n})0!\\ & =\sum _{i=1}^n(-1{)}^{i+1}(\genfrac{}{}{0ex}{}{n}{i})(n-i)!\\ & =n!\sum _{i=1}^n\frac{(-1{)}^{i+1}}{i!},\end{array}}$$ and since a derangement is a permutation that leaves none of the n objects fixed, this implies $${\displaystyle D_n=n!-\left|S_1\cup \cdots \cup S_n\right|=n!\sum _{i=0}^n\frac{(-1{)}^i}{i!}.}$$

On the other hand, $${\displaystyle n!=\sum _{i=0}^n(\genfrac{}{}{0ex}{}{n}{i})D_i}$$ since we can choose n − i elements to be in their own place and derange the other i elements in just D_i ways, by definition.

Growth rate

From $${\displaystyle D_n=n!\sum _{i=0}^n\frac{(-1{)}^i}{i!}}$$ and $${\displaystyle e^x=\sum _{i=0}^{\mathrm{\infty }}\frac{x^i}{i!}}$$ by substituting x = −1 one immediately obtains that $${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{D_n}{n!}=\underset{n\to \mathrm{\infty }}{lim}\sum _{i=0}^n\frac{(-1{)}^i}{i!}=e^{-1}\approx 0.367879\dots .}$$ 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 D_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.

Asymptotic expansion in terms of Bell numbers

An asymptotic expansion for the number of derangements in terms of Bell numbers is as follows: $${\displaystyle D_n=\frac{n!}{e}+\sum _{k=1}^m{\left(-1\right)}^{n+k-1}\frac{B_k}{n^k}+O\left(\frac{1}{n^{m+1}}\right),}$$ where m is any fixed positive integer, and B_k denotes the k th Bell number. Moreover, the constant implied by the big O-term does not exceed B_{m + 1}.

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 (f, g) such that f is in U and g is in V, and for all a ∈ 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 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 (after a proper use of the inclusion-exclusion formula) that the answer has the form $${\displaystyle {\int }_0^{\mathrm{\infty }}{P}_{n_1}(x)P_{n_2}(x)\cdots P_{n_r}(x)e^{-x}dx,}$$ 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 are the Laguerre polynomials (up to a sign that is easily decided).

In particular, for the classical derangements, one has that $${\displaystyle D_n=\frac{\mathrm{\Gamma }(n+1,-1)}{e}={\int }_0^{\mathrm{\infty }}(x-1{)}^n{e}^{-x}dx}$$ where Γ(s, x) is the upper incomplete gamma function.

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.