A bijection on Dyck paths and its consequences
A Dyck path of semilength \(n\) is a path in the first quadrant beginning at \((0,0)\), ending at \((2n,0)\), and consisting of rises (steps of \((1,1)\)) and falls (steps of \((1,-1)\)). This article gives several proofs, one of which uses a bijection, of the fact that there are as many Dyck paths of semilength \(n\) beginning with exactly \(k\) consecutive rises (height of the first peak equals \(k\)) as there are with exactly \(k\) falls landing on the horizontal axis (\(k\) returns). As a consequence of the bijection it is shown that the Narayana numbers \(u(n,k)= \left({1\over n}\right) \left(\begin{smallmatrix} n\\ k\end{smallmatrix}\right) \left(\begin{smallmatrix} n\\ k+1\end{smallmatrix}\right)\) (*), which count Dyck paths of semilength \(n\) with \(k\) valleys (falls followed by rises) also count Dyck paths of semilength \(n\) with \(k\) high peaks (rises to a height of at least 2 followed by falls). (*) Reviewer's remark: It would have been helpful if the article had stated which of the references contains this result and had included the figure to which it refers.
scientific article; zbMATH DE number 1109999