Grand Dyck paths with air pockets

TL;DR

This paper studies a generalized class of Dyck paths called GDAP, exploring their enumeration under various constraints and establishing connections with other combinatorial structures.

Contribution

It introduces GDAP as a new generalization of Dyck paths with air pockets and provides enumeration results and bijections with known combinatorial classes.

Findings

Enumerative formulas for GDAP with height restrictions

Bijections with other combinatorial classes in special cases

Analysis of path properties like last point and first return

Abstract

Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths with air pockets by allowing them to go below the $x$-axis. We present enumerative results on GDAP (or their prefixes) subject to various restrictions such as maximal/minimal height, ordinate of the last point and particular first return decomposition. In some special cases we give bijections with other known combinatorial classes.