### Lemma

The number of permutations of with **k** descents is equal to
the number with **k** drops, i.e.,

### Proof

A descent of must lie inside a cycle of since our conventions
guarantee that the last element in a cycle is followed by a larger integer.
By the meaning of the cycle decomposition (namely, that elements within cycles are mapped to the next element in the cycle) we see that a descent of
corresponds to a drop of .
Conversely, a drop in must occur within a cycle (i.e., not in passing from the last
element of a cycle to the first) and corresponds to a descent in .
Thus the number of permutations with **k** descents is equal to the number with **k** drops.