A human proof for a generalization of Shalosh B. Ekhad's 10n Lattice Paths Theorem

Consider lattice paths in Z taking unit steps north (N) and east (E). Fix positive integers r, s and put an equivalence relation on points of Z by letting v, w be equivalent if v − w = `(r, s) for some ` ∈ Z. Call a lattice path valid if whenever it enters a point v with an E-step, then any further points of the path in the equivalence class of v are also entered with an E-step. Loehr and Warrington conjectured that the number of valid paths from (0, 0) to (nr, ns) is ` r+s r ́ n . We prove this conjecture when s = 2.