Derangements arise in a number of guises in combinatorial problems. For example, each solution to the rooks problem, where n rooks must be placed on an n x n chessboard such that no two rooks occupy the same row or column, can be considered as a derangment of n elements. 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.
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 [1,n], then for some k in [1,n-1], φn(n) = k. Then if we let (k,n) be the permutation of [1,n] which swaps k and n, and we let φn-1 be the composition ((k,n) o φn); then φn-1(n) = n, and either:
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)56The 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 -> 315624Thus, if we write dn as the number of derangements of n letters, and we define d0 = 1, d1 = 0; then dn satisfies the recurrence:
Perhaps a more well-known method of counting derangements uses the inclusion-exclusion principle.
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)).
Generalizations
External Links