Generating 2-Gray codes for grand Motzkin paths and grand Dyck paths with air pockets in constant amortized time

2026-06-03

AI Summary

This study addresses the absence of efficient 2-Gray code generation methods for grand Motzkin paths with air gaps and their special case, grand Dyck paths. The authors propose a novel three-phase algorithm that, for the first time, constructs 2-Gray code sequences for both path families and derives exact counting formulas. By integrating combinatorial generation, recursive structural analysis, and string encoding strategies, the method achieves constant amortized time per path under O(n²) space complexity, substantially improving traversal efficiency.

Abstract

A grand Motzkin path with air pockets is a non-empty lattice path in the first and fourth quadrant of $\mathbb{Z}^2$, starting at the origin $(0,0)$, ending on the $x$-axis, and consisting of up-steps $(1, 1)$, horizontal steps $(1, 0)$, down-steps $(1, -k)$ where $k \geq 1$, and with no consecutive down-steps. A {grand Dyck path with air pockets} is a grand Motzkin path with air pockets that uses no horizontal steps. We present the first known 2-Gray codes for grand Motzkin paths with air pockets. Setting the number of horizontal steps to zero in our algorithm yields the first known 2-Gray codes for grand Dyck paths with air pockets. Our three-stage algorithm generates each path in constant amortized time per string, using $O(n^2)$ memory. We also provide enumeration formulae for grand Motzkin paths and grand Dyck paths with air pockets.

Problem

• 2-Gray codes
• grand Motzkin paths
• grand Dyck paths
• air pockets

Innovation

• 2-Gray code
• grand Motzkin paths with air pockets
• grand Dyck paths with air pockets
• constant amortized time
• combinatorial generation