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

  • Michael Z. Spivey
Keywords: Lattice path, bijection


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.
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