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

Abstract

We provide a direct geometric bijection for the number of lattice paths that never go below the line y ¼ 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 ¼ 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. r

The number of lattice paths below a cyclically shifting boundary

“…In fact, our proof is a generalization of the bijection used in [6] to prove Corollary 3. (Our bijection reduces to that of [6] in the case when all parts of a are the same, though an allowance must be made for the cyclically shifting boundary.) Section 3 contains a brief account of an alternative derivation of Theorem 1 using the cycle lemma.…”

“…We do so by showing that bad paths are in bijection with a less restrictive set of paths, in the spirit of André's reflection principle [1]. In fact, our proof is a generalization of the bijection used in [6] to prove Corollary 3. (Our bijection reduces to that of [6] in the case when all parts of a are the same, though an allowance must be made for the cyclically shifting boundary.)…”

“…Notice that in each of the top three rows of the figure there are a total of 36 dominated paths from (0, 0) to (6,3). There are this many also in the bottom row if the three identical cyclic shifts of a = (2, 2, 2) are taken into account.…”

Counting Lattice Paths via a New Cycle Lemma

“…In the case α is a positive integer and β = 1, the following result is given as Theorem 3.4.3 of [7](see also Theorem 7 of [3]). …”

“…Before closing the section, we remark that the formula (1) is proved and generalized in [3] by using the reflection method instead of the cycle method. Generalizations of the formula (1) are also seen in [5].…”