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.
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1968 ◽
Vol 11
(4)
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pp. 537-545
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1948 ◽
Vol os-19
(1)
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pp. 238-248
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1966 ◽
Vol 1
(2)
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pp. 224-232
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