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

number of returns to the axis $RET(D)$
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?

flag	asked May 18 2011 at 19:53 Martin Rubey 1,447●5●13

Gjergji Zaimi · Accepted Answer · 2011-05-18 20:57:32Z UTC

0

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)

edited May 18 2011 at 20:57

answered May 18 2011 at 20:44

Gjergji Zaimi
37.5k●2●72●149

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 2011 at 9:42

Equidistribution of returns and height of first peak of Dyck paths

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