Shortest-path diagrams

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Here are some diagrams that represent the possible paths of length 2n from one corner of an n-by-n grid to the opposite corner. The number of paths are the central binomial coefficients

or ,

central meaning they fall along the center line of Pascal's triangle.

(Oddly enough, the Catalan numbers describe how many of these paths stay under the main diagonal.)

1 x 1 grid, 2 paths:

2 x 2 grid, 6 paths:

3 x 3 grid, 20 paths:

4 x 4 grid, 70 paths:

5 x 5 grid, 252 paths:

Designed and rendered using Mathematica 3.0 for the Apple Macintosh.

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