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, 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 inĀ 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.