Catalan Traffic at the Beach
نویسنده
چکیده
The ubiquitous Catalan numbers Cn = (2n n ) / (n + 1) occur as t (n,−n) in the following table showing the number of ways to reach the point (n,m) on a rectangular grid under certain traffic restrictions, indicated by arrows. m 1↑→ 3↑→ 6↑→ 9↑→ 12↑→ 12↑→ 12↓→ ↘ 2 1↑→ 2↑→ 3↑→ 3↑→ 3↓→ ↘ 12↓→↘ 1 1↑→ 1↑→ 1↓→ ↘ 3↓→↘ 3↓→↘ 12↓→↘ start here 1↑→gate ↘ 1↓→↘ 1↓→↘ 3↓→↘ 6↓→↘ 15↓→↘ −1 ≈ 1 →↘ 1↓→↘ 2↓→↘ 4↓→↘ 9↓→↘ 21↓→↘ −2 ≈ ≈ 2 →↘ 3↓→↘ 6↓→↘ 13↓→↘ 30↓→↘ −3 ≈ ≈ ≈ 5 →↘ 9↓→↘ 19↓→↘ 43↓→↘ −4 ≈ t (n,m) ≈ 14 →↘ 28↓→↘ 62↓→↘ −5 ≈ ≈ ≈ ≈ ≈ 42 →↘ 90↓→↘ 0 1 2 3 4 5 n We prove this with the help of hypergeometric identities, and also by solving an equivalent lattice path problem. On the way we pick up several identities and discuss other known sequences of numbers occurring in the Catalan traffic scheme, like the Motzkin numbers in row m = −1, and the “Tri-Catalan numbers” 1, 1, 3, 12, 55, . . . at the gates.
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ورودعنوان ژورنال:
- Electr. J. Comb.
دوره 9 شماره
صفحات -
تاریخ انتشار 2002