# FindStatFact 2: Weak exceedances on Dyck paths

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 $\pi$ to the left-to-right maxima of $\phi(\pi)$, then
• sending $\phi(\pi)$ to its increasing tree $T(\phi(\pi))$, and finally
• sending $T(\phi(\pi))$ to the Dyck path $D$ by identifying an up step in $D$ with a left subtree in $T(\phi(\pi))$, and a down step in $D$ with a right subtree in $T(\phi(\pi))$,

will send the number of weak exceedances of $\pi$ to the number of peaks of a Dyck path.