Lattice paths and the diagonal of the cube

Abstract: We study lattice paths in the cube, starting at $(n,n,n)$ and ending at $(0,0,0)$, with unit steps $(-1,0,0)$, $(0,-1,0)$, $(0,0,-1)$. Our main interest is the number of times the diagonal $x=y=z$ is visited during the random walk. We derive the corresponding generating function of such lattice paths. We also turn to the cuboid, enumerating lattice paths starting at $(n_1,n_2,n_3)$ and ending at $(0,0,0)$ according to the visits to the cube's diagonal. Furthermore, we provide for the cube a refined enumeration according to visits after a detour of a certain length. These enumerations allow us to obtain distributional results for the corresponding random variables. Extensions to hypercube are discussed, as well as a summary of known results for the square. We collect applications to the sampling without replacement urn and a card guessing game. Finally, we also show how to recover (and extend) a very recent result of Li and Starr on Dyck bridges using generating functions and composition schemes.