Random Permutation Statistics - Number of Permutations Containing Cycles

Number of Permutations Containing Cycles

Applying the Flajolet–Sedgewick fundamental theorem, i.e. the labelled enumeration theorem with, to the set

we obtain the generating function

g_m(z) = \frac{1}{|S_m|} \left( \log \frac{1}{1-z} \right)^m = \frac{1}{m!} \left( \log \frac{1}{1-z} \right)^m.

The term

(-1)^{n+m} n! \; g_m(z) = \left

yields the Stirling numbers of the first kind, i.e. is the EGF of the unsigned Stirling numbers of the first kind.

We can compute the OGF of these numbers for n fixed, i.e.

s_n(w) = \sum_{m=0}^n \left w^m.

Start with

g_m(z) = \sum_{n\ge m} \frac{(-1)^{n+m}}{n!} \left z^n

which yields

(-1)^m g_m(z) w^m = \sum_{n\ge m} \frac{(-1)^n}{n!} \left w^m z^n.

Summing this, we obtain

\sum_{m\ge 0} (-1)^m g_m(z) w^m = \sum_{m\ge 0} \sum_{n\ge m} \frac{(-1)^n}{n!} \left w^m z^n = \sum_{n\ge 0}\frac{(-1)^n}{n!} z^n \sum_{m=0}^n \left w^m.

Using the formula for on the left, the definition of on the right, and the binomial theorem, we obtain

(1-z)^w = \sum_{n\ge 0} {w \choose n} (-1)^n z^n = \sum_{n\ge 0}\frac{(-1)^n}{n!} s_n(w) z^n.

Comparing the coefficients of, and using the definition of the binomial coefficient, we finally have

a falling factorial.

Read more about this topic:  Random Permutation Statistics

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