Dyck paths on black-and-white lattices

Abstract

A Dyck path of semilength $n$ is a lattice path from $(0,0)$ to $(n,n)$ consisting of $n$ right-steps $(1,0)$ and $n$ up-steps $(0,1)$ that never rises above the line $y=x$. These paths are enumerated by the Catalan numbers and play a central role in enumerative combinatorics. We color the cells of the integer grid in black and white according to two natural patterns, namely chessboard and column-alternating, and enumerate the Dyck paths having equal numbers of black and white cells beneath them.