What's the number that follows in this sequence, and what is the sequence's significance?

0,1,2,9,44,265,...

The next number in the series is 1854. The series is generated by

a(n)=n!*∑[(-1^x)/x!] for x=0 to n

What this series represents is the number of permutations of a set, such that none of the members of the permutation are in the same position as the members of the original set. To illustrate:

{A}: There are no permutations that meet the requirement (0 instances)

{A,B}: {B,A} (1 instance)

{A,B,C}: {B,C,A}, {C,A,B} (2 instances)

{A,B,C,D}: {B,A,D,C}, {B,C,D,A}, {B,D,A,C}, {C,A,D,B}, {C,D,A,B}, {C,D,B,A}, {D,A,B,C}, {D,C,A,B}, {D,C,B,A} (9 instances)

and so on.

Posted by Bryan on 2003-05-12 11:35:25