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

 Binomial[2n, n] or  (2n)!/(n!)^2 ,

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

The first few are 1, 2, 6, 20, 70, 252, ...

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

1 x 1 grid, 2 paths:
[ 1 x 1 (sorry, but the pictures speak a thousand words) ]

2 x 2 grid, 6 paths:
[ 2 x 2 ]

3 x 3 grid, 20 paths:
[ 3 x 3 ]

4 x 4 grid, 70 paths:
[ 4 x 4 ]

5 x 5 grid, 252 paths:
[ 5 x 5 ]

Designed and rendered using Mathematica 3.0 for the Apple Macintosh.

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