Delannoy numbers describe the number of paths from the southwest corner of a rectangular grid to the northeast corner, using only single steps north, northeast, or east. Delannoy numbers can be computed recursively using this formula:

*D*(*a*, *b*) =
*D*(*a*−1, *b*) + *D*(*a*, *b*−1) +
*D*(*a*−1, *b*−1) [Weisstein 1999],

where *D*(0, 0) = 1.

For a 1 × 1 grid, there are 3 paths:

For a 2 × 2 grid, there are 13 paths:

3 × 3 grid, 63 paths:

The Delannoy paths that do not rise above the SW–NE diagonal (the paths drawn above in green)
represent the *Schröder numbers*.

Another interpretation of the Schröder numbers is the number of ways a rectangle containing *n* points—with no two points falling on a line parallel to the rectangle’s edges—can be sliced into smaller rectangles, where each slice intersects one of the points, is parallel to one of the rectangle’s edges, and divides only a single rectangle.

1 slice, 2 ways:

2 slices, 6 ways:

3 slices, 22 ways:

4 slices, 90 ways:

See also “Schröder Rectangulations” at the Wolfram Demonstrations Project.

The Motzkin numbers describe the number of paths from the southwest corner of a grid to
the southeast corner, using only steps northeast, east, and southeast. (Clearly, for a grid
with width *n* the maximum necessary height is ⌊*n*/2⌋.)

2 × 2 grid, 2 paths:

3 × 3 grid, 4 paths:

4 × 4 grid, 9 paths:

5 × 5 grid, 21 paths:

Definitions and formulas cribbed from Eric Weisstein’s CRC Concise Encyclopedia of Mathematics (CRC Press, 1999), pp. 411 and 1201.

Figures originally designed and rendered using
*Mathematica*.

You might also enjoy Counting Lattice Paths.

© 1999–2023 Robert Dickau.

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