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

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.