Maintaining the spirit of the reflection principle when the boundary has arbitrary integer slope

We provide a direct geometric bijection for the number of lattice paths that never go below the line y = kx for a positive integer k. This solution to the Generalized Ballot Problem is in the spirit of the reflection principle for the Ballot Problem (the case k = 1), but it uses rotation instead of reflection. It also gives bijective proofs of the refinements of the Generalized Ballot Problem which consider a fixed number of right-up or up-right corners. 1 The Classical Ballot Problem A lattice path is a path in the plane consisting of unit up-steps and right-steps, whose ends are points with integer coordinates. The classical Ballot Problem was given in Bertrand [3]: Theorem 1 For n ≥ m ≥ 0, the number of lattice paths from (0, 0) to (m,n) that never go below the diagonal y = x is n−m+ 1 n+ 1 (