Retakh's Motzkin paths and some combinatorial comments

TL;DR

This paper explores a special class of Dyck paths with peaks restricted to certain levels, connecting their enumeration to Motzkin numbers and analyzing their average height, with additional combinatorial insights.

Contribution

It introduces combinatorial comments on the enumeration of these paths, linking them to Motzkin numbers and providing average height analysis.

Findings

Enumeration involves Motzkin numbers

Average height of paths analyzed

Provides combinatorial comments on path structure

Abstract

Dyck paths where peaks are only allowed on level 1 and on even-indexed levels, were introduced by Retakh and analysed by Zeilberger, with assistance from Ekhad. We add some combinatorial comments to the enumeration, which involves Motzkin numbers, in particular, about the average height of such objects.