Lattice ball

A lattice-ball is a set of lattice points which are close to the origin. They can be viewed as the lattice points contained in a cross-polytope. They are counted by the Delannoy numbers.

Definition

Fix any positive integer $n$ and nonnegative integer $m$. Let $\Vert \cdot {\Vert }_1$ denote the Manhattan norm. Then the lattice ball of radius $m$ in ${\mathbb{\mathbb{Z}}}^n$ is defined as the set of all points $p\in {\mathbb{\mathbb{Z}}}^n$ such that

$\Vert p{\Vert }_1\le m$

Counting Points

The Delannoy number $d_{m,n}$ counts the number of points inside the lattice ball of radius $m$ in ${\mathbb{\mathbb{Z}}}^n$.

See Also

• Combinatorics

References

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