A permutation of distinct, ordered items in which none of the items is in its original ordered position is known as a derangement. If some, but not necessarily all, of the items are not in their original ordered positions, the configuration can be referred to as a partial derangement (Evans et al. 2002, p. 385).

Among the possible permutations of distinct items, examine the number of these permutations in which exactly items are in their original ordered positions. Then

			(1)
			(2)
			(3)

where is a binomial coefficient and is the subfactorial

Here is a table of the number of partial derangements for , 1, ..., 8:

(4)

(Sloane's A098825).

This entry contributed by Gerald Del Fiacco

Evans, C. D. H.; Hughes, J.; and Houston, J. "Significance-Testing the Validity of Idiographic Methods: A Little Derangement Goes a Long Way." Brit. J. Math. Stat. Psych. 55, 385-390, 2002.

Skiena, S. "Derangements." §1.4.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 33-34, 1990.

Sloane, N. J. A. Sequence A098825 in "The On-Line Encyclopedia of Integer Sequences."

CITE THIS AS:

Fiacco, Gerald Del. "Partial Derangement." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/PartialDerangement.html