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:

“I call it our collective inheritance of isolation. We inherit isolation in the bones of our lives. It is passed on to us as sure as the shape of our noses and the length of our legs. When we are young, we are taught to keep to ourselves for reasons we may not yet understand. As we grow up we become the “men who never cry” and the “women who never complain.” We become another generation of people expected not to bother others with our problems.”
—Paula C. Lowe (20th century)

“To finish the moment, to find the journey’s end in every step of the road, to live the greatest number of good hours, is wisdom. It is not the part of men, but of fanatics, or of mathematicians, if you will, to say, that, the shortness of life considered, it is not worth caring whether for so short a duration we were sprawling in want, or sitting high. Since our office is with moments, let us husband them.”
—Ralph Waldo Emerson (1803–1882)

“Man always made, and still makes, grotesque blunders in selecting and measuring forces, taken at random from the heap, but he never made a mistake in the value he set on the whole, which he symbolized as unity and worshipped as God. To this day, his attitude towards it has never changed, though science can no longer give to force a name.”
—Henry Brooks Adams (1838–1918)