Stirling Number of the First Kind

The Stirling number of the first kind
is the number of Permutation group comprised of cycles.

Theorem

Proof

is either a fixpoint or not

Theorem

Proof 1

The coefficient of in
which we write satisfies:

So it satisfies the same recurrence relation as
and check some initial conditions.

Proof 2

Show

by counting cycles.

Proof 3

Use Burnside’s Lemma.
Let act on by permuting the domain

Theorem