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## Saturday, May 1, 2010

### Fixed-point-free permutations

The permuations of the elements of {1,2,3} are followed by the permutioncycle(s)
1-2-3 = (1)(2)(3)
1-3-2 = (1)(23)
2-1-3 = (12)(3)
2-3-1 = (123)
3-1-2 = (132)
3-2-1 = (13)(2).
Only two of them are so called "fixed-point-free", i.e. they have no cycles of length 1: 2-3-1 and 3-1-2. There is a symbol for this number D for derangement. So D(3)=2. We can easily find D(4) by thinking in permutation cycles: those that do not include length 1 cycles are fixed-point-free. The possible cycle structures of permutations of length 4 are 1-to-1 related to the partitions of 4 ( number of permutations ).
4 = xxxx ( 6 = 4*3*2*1 / 4 )
31= xxx-x ( 8 = 4*3*2 / 3 )
22 = xx-xx ( 3 = 4*3 / 2 * 2 )
211 = xx-x-x ( 6 = 4*3 /2)
1111 = x-x-x-x ( 1 = 1 )
All basic M208/GTA1 material, I suppose. This formula is however not in the course.
$$D(n) = n! \cdot \sum_{k=0}^{n} \frac{(-1)^k}{k!}$$
With the formula for D we can simply check the result for 4.
$$D(4) = 4! \cdot ( \frac{(-1)^0}{0!} + \frac{(-1)^1}{1!} + \frac{(-1)^2}{2!} + \frac{(-1)^3}{3!} + \frac{(-1)^4}{4!} ) =9.$$

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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)