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.