Derangement
2020 Mathematics Subject Classification: Primary: 05A15 [MSN][ZBL]
A permutation of 
elements in which the element 
cannot occupy the 
-th position, 
. The problem of calculating the number 
of derangements is known as the "Montmort matching problem" or "problème des rencontres" (cf. Classical combinatorial problems). The following formula holds:
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Derangements are a special case of permutations satisfying a specific restriction on the position of the permuted elements. For example, the "problème des ménages" consists in calculating the number 
of permutations conflicting with the two permutations 
and 
. (Two permutations of 
elements are called conflicting if the 
-th element occupies different positions in each of them for all 
). The number 
is given by the formula:
 |
The number 
of Latin squares (cf. Latin square) of size 
for 
can be calculated in terms of 
and 
by the formulas
 |
 |
References
| [1] | H.J. Ryser, "Combinatorial mathematics" , Carus Math. Monogr. , 14 , Wiley & Math. Assoc. Amer. (1963) |
| [2] | J. Riordan, "An introduction to combinatorial mathematics" , Wiley (1958) |
Derangement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derangement&oldid=39888