Enumeration of strings in Dyck paths: A bijective approach

Left to right maxima in Dyck Paths

“…For a detailed study of properties of Dyck paths see [6]. For further recent work on Dyck paths, see [1,2,3,4,5,8,14].…”

A Bijection on Bilateral Dyck Paths

“…Note that the restriction of our bijection to Dyck paths is essentially identical to a bijection given in [6]. There, the authors used two length-preserving bijections on Dyck paths to show the equidistribution of statistics of certain strings occurring at odd height, at even height, and anywhere.…”

“…There, the authors used two length-preserving bijections on Dyck paths to show the equidistribution of statistics of certain strings occurring at odd height, at even height, and anywhere. However, the problem discussed here, regarding the equidistribution of up-steps at odd height and peaks anywhere, was not addressed in [6].…”

General Results on the Enumeration of Strings in Dyck Paths

“…Many articles dealing with the occurrence of strings in Dyck paths have appeared in the literature (e.g. see [1,3,5,8,12,13,14,19,20,21,24]). In particular, it has been proved (see [8]) that the statistic N τ follows the Narayana distribution (A001263 of [23]), for every string τ of length 2, the statistic N udu follows the Donaghey distribution (see [24]) and the statistic N duu follows the Touchard distribution (see [8]).…”