無

In this thesis, our goal is to use mathematical induction to give a direct proof to show that the number of b(n - m, k ; n, k - m) is m/(n+k-m) ,where b(n – m, k; n, k - m) denotes the number of non-intersecting paths that the upper path goes from (0, 0) to(n - m, k) while the lower path goes from (0, 0) to (n, k - m). Furthermore, we conclude two applications about b(n-m, k ; n, k-m), namely b(n, k) (see Definition 2.2) and PP(n, k) (see Definition 4.4). We also bring up some open problems concerning our topics.

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