Random Permutation Statistics - Number of Permutations Containing An Even Number of Even Cycles

Number of Permutations Containing An Even Number of Even Cycles

We may use the Flajolet–Sedgewick fundamental theorem directly and compute more advanced permutation statistics. (Check that page for an explanation of how the operators we will use are computed.) For example, the set of permutations containing an even number of even cycles is given by

$\mathfrak{P}(\mathfrak{C}_{\operatorname{odd}}(\mathcal{Z})) \mathfrak{P}_{\operatorname{even}}(\mathfrak{C}_{\operatorname{even}}(\mathcal{Z})).$

Translating to exponential generating functions (EGFs), we obtain

$\exp \left( \frac{1}{2} \log \frac{1+z}{1-z} \right) \cosh \left( \frac{1}{2} \log \frac{1}{1-z^2} \right)$

$\frac{1}{2} \exp \left( \frac{1}{2} \left( \log \frac{1+z}{1-z} + \log \frac{1}{1-z^2} \right) \right) + \frac{1}{2} \exp \left( \frac{1}{2} \left( \log \frac{1+z}{1-z} - \log \frac{1}{1-z^2} \right) \right).$

This simplifies to

$\frac{1}{2} \exp \left( \frac{1}{2} \log \frac{1}{(1-z)^2} \right) + \frac{1}{2} \exp \left( \frac{1}{2} \log (1+z)^2 \right)$

$\frac{1}{2} \frac{1}{1-z} + \frac{1}{2} (1+z) = 1 + z + \frac{1}{2} \frac{z^2}{1-z}.$

This says that there is one permutation of size zero containing an even number of even cycles (the empty permutation, which contains zero cycles of even length), one such permutation of size one (the fixed point, which also contains zero cycles of even length), and that for, there are such permutations.

Read more about this topic: Random Permutation Statistics

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

“I am walking over hot coals suspended over a deep pit at the bottom of which are a large number of vipers baring their fangs.”
—John Major (b. 1943)

“Of all reformers Mr. Sentiment is the most powerful. It is incredible the number of evil practices he has put down: it is to be feared he will soon lack subjects, and that when he has made the working classes comfortable, and got bitter beer into proper-sized pint bottles, there will be nothing left for him to do.”
—Anthony Trollope (1815–1882)

“The new shopping malls make possible the synthesis of all consumer activities, not least of which are shopping, flirting with objects, idle wandering, and all the permutations of these.”
—Jean Baudrillard (b. 1929)

“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)