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.

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