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I believe that it is "well known" that the following two statistics on Dyck paths have symmetric joint distribution:

  1. number of returns to the axis $RET(D)$
  2. height of the first peak (or length of the last descent) $HFP(D)$

That is: $\sum_{D} x^{RET(D)}y^{HFP(D)} = \sum_{D} x^{HFP(D)}y^{RET(D)}$

However, I could not find a reference for that. Might it be due to Kreweras?

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up vote 1 down vote accepted

You can use the article "A bijection on Dyck paths and its consequences" by E. Deutsch. The author has several papers on enumerative problems on Dyck/Motzkin paths. (See also here)

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Many thanks! I'll leave the question open for another day just in case somebody comes up with an earlier reference... – Martin Rubey May 19 '11 at 9:42

For future references: as we have discussed in this question, this also follows from the "zeta map" sending the bistatistic (area,bounce) to the bistatistic (dinv,area). For another definition and further background, see also page 50 of Jim Haglund's book.

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