Tags: #theorem

Statement

The number of exceedances is equidistributed to the number of descents. In particular, this means that the former is Eulerian statistic.

Proof

We note that exceedance is equidistributed to antiexceedance and build a bijection there. We shall use the same map as in records is Sterlingian, so $\mathrm{aexc}(w)=\mathrm{des}(f(w))$.

Then, we note that whenever there is an antiexceedance, in cycle notation, this is where a bigger number is sent to a smaller number (so bigger number followed by a smaller number). When we convert it with the $f$ map, this becomes a descent, and vice versa.