Definition 5.6.4
is the number of permutations of
with precisely
cycles (including fixed points).

For instance
,
,
,
for all
but
.

Lemma 5.6.5

Proof.
Either the point is in a cycle on its own (
such) or it
is not. In this case, can be inserted into any of places in
any of the permutations of
.

We can use this recurrence to prove this proposition. (Proof left as exercise.)