Random Permutation Statistics - Derangements Containing An Even and An Odd Number of Cycles

Derangements Containing An Even and An Odd Number of Cycles

We can use the same construction as in the previous section to compute the number of derangements containing an even number of cycles and the number containing an odd number of cycles. To do this we need to mark all cycles and subtract fixed points, giving

$g(z, u) = \exp\left( - u z + u \log \frac{1}{1-z} \right) = \exp(-uz) \left( \frac{1}{1-z} \right)^u.$

Now some very basic reasoning shows that the EGF of is given by

$q(z) = \frac{1}{2} \times g(z, -1) + \frac{1}{2} \times g(z, 1) = \frac{1}{2} \exp(-z) \frac{1}{1-z} +\frac{1}{2} \exp(z) (1-z).$

We thus have

$D_0(n) = n! q(z) = \frac{1}{2} n! \sum_{k=0}^n \frac{(-1)^k}{k!} + \frac{1}{2} n! \frac{1}{n!} - \frac{1}{2} n! \frac{1}{(n-1)!}$

which is

$\frac{1}{2} n! \sum_{k=0}^n \frac{(-1)^k}{k!} + \frac{1}{2} (1-n) \sim \frac{1}{2e} n! + \frac{1}{2} (1-n).$

Subtracting from, we find

The difference of these two ( and ) is

Read more about this topic: Random Permutation Statistics

Famous quotes containing the words odd, number and/or cycles:

“It’s a very odd thing—
As odd as can be—
That whatever Miss T. eats
Turns into Miss T.;”
—Walter De La Mare (1873–1956)

“It seems to me that there must be an ecological limit to the number of paper pushers the earth can sustain, and that human civilization will collapse when the number of, say, tax lawyers exceeds the world’s total population of farmers, weavers, fisherpersons, and pediatric nurses.”
—Barbara Ehrenreich (b. 1941)

“The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
—Linda Goodman (b. 1929)