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

AbstractWe 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