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?



Equidistribution of returns and height of first peak of Dyck paths
You can use the article ["A bijection on Dyck paths and its consequences"](https://doi.org/10.1016/S0012-365X(97)00097-6 "Discrete Math. 179, No. 1-3, 253-256 (1998); corrigendum ibid. 187, No. 1-3, 297 (1998). zbMATH review at https://zbmath.org/0890.05005") by E. Deutsch. The author has several papers on enumerative problems on Dyck/Motzkin paths. (See also [here](https://doi.org/10.1016/S0012-365X(98)00371-9 "Deutsch, Emeric. Dyck path enumeration. Discrete Math. 204, No. 1-3, 167-202 (1999). zbMATH review at https://zbmath.org/0932.05006"))


