Random Permutation Statistics - Expected Number of Transpositions of A Random Permutation | Expected Number Transpositions Random Permutation

Expected Number of Transpositions of A Random Permutation

We can use the disjoint cycle decomposition of a permutation to factorize it as a product of transpositions by replacing a cycle of length k by transpositions. E.g. the cycle factors as . The function for cycles is equal to and we obtain

$g(z, u) = \left( \frac{1}{1-uz} \right)^{1/u}$

and

$\frac{\partial}{\partial u} g(z, u) \Bigg|_{u=1} = \frac{1}{1-z} \sum_{k\ge 1} (k-1) \frac{z^k}{k} = \frac{z}{(1-z)^2} - \frac{1}{1-z} \log \frac{1}{1-z}.$

Hence the expected number of transpositions is

We could also have obtained this formula by noting that the number of transpositions is obtained by adding the lengths of all cycles (which gives n) and subtracting one for every cycle (which gives by the previous section).

Note that again generates the unsigned Stirling numbers of the first kind, but in reverse order. More precisely, we have

$(-1)^m n! \; g(z, u) = \left$

To see this, note that the above is equivalent to

$(-1)^{n+m} n! \; g(z, u)|_{u=1/u} |_{z=uz} = \left$

and that

$g(z, u)|_{u=1/u} |_{z=uz} = \left( \frac{1}{1-z} \right)^u = \frac{1}{m!} \left( \log \frac{1}{1-z} \right)^m,$

which we saw to be the EGF of the unsigned Stirling numbers of the first kind in the section on permutations consisting of precisely m cycles.

Read more about this topic: Random Permutation Statistics

Famous quotes containing the words expected, number and/or random:

“[My father] was a lazy man. It was the days of independent incomes, and if you had an independent income you didn’t work. You weren’t expected to. I strongly suspect that my father would not have been particularly good at working anyway. He left our house in Torquay every morning and went to his club. He returned, in a cab, for lunch, and in the afternoon went back to the club, played whist all afternoon, and returned to the house in time to dress for dinner.”
—Agatha Christie (1891–1976)

“... in every State there are more women who can read and write than the whole number of illiterate male voters; more white women who can read and write than all Negro voters; more American women who can read and write than all foreign voters.”
—National Woman Suffrage Association. As quoted in History of Woman Suffrage, vol. 4, ch. 13, by Susan B. Anthony and Ida Husted Harper (1902)

“It is a secret from nobody that the famous random event is most likely to arise from those parts of the world where the old adage “There is no alternative to victory” retains a high degree of plausibility.”
—Hannah Arendt (1906–1975)