Random Permutation Statistics - Expected Number of Transpositions of A 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:

    “Ancient history has an air of antiquity. It should be more modern. It is written as if the specator should be thinking of the backside of the picture on the wall, or as if the author expected that the dead would be his readers, and wished to detail to them their own experience.”
    —Henry David Thoreau (1817–1862)

    “In view of the fact that the number of people living too long has risen catastrophically and still continues to rise.... Question: Must we live as long as modern medicine enables us to?... We control our entry into life, it is time we began to control our exit.”
    —Max Frisch (1911–1991)

    “And catch the gleaming of a random light,
    That tells me that the ship I seek is passing, passing.”
    —Paul Laurence Dunbar (1872–1906)