Who knows of a way to describe the number of weak exceedances of a permutation in terms of Dyck paths?
Go to http://www.findstat.org/St000213, click on search for statistic, and you will find out that
- applying the inverse of the first fundamental transformation $\latex \phi$ given by sending the maximal entries in the cycles of a permutation to the left-to-right maxima of , then
- sending $\phi(\pi)$ to its increasing tree , and finally
- sending to the Dyck path by identifying an up step in with a left subtree in , and a down step in with a right subtree in ,
will send the number of weak exceedances of to the number of peaks of a Dyck path.