Enumerating Lattice Paths Touching or Crossing the Diagonal at a Given Number of Lattice Points

We give bijective proofs that, when combined with one of the combinatorial proofs of the general ballot formula, constitute a combinatorial argument yielding the number of lattice paths from (0,0) to (n,rn) that touch or cross the diagonal y=rx at exactly k lattice points. This enumeration partitions all lattice paths from (0,0) to (n,rn). While the resulting formula can be derived using results from Niederhausen, the bijections and combinatorial proof are new