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

Loehr, Nicholas A.; Sagan, Bruce E.; Warrington, Gregory S.

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

Authors

Loehr, Nicholas A.

Sagan, Bruce E.

Warrington, Gregory S.

Description

Consider lattice paths in Z^2 taking unit steps north (N) and east (E). Fix positive integers r,s and put an equivalence relation on points of Z^2 by letting v,w be equivalent if v - w = m (r,s) for some m in 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 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 choose r)^n. We prove this conjecture when s = 2.
9 pages, 6 figures