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 $\|\cdot \|_{1}$ denote the Manhattan norm. Then the lattice ball of radius $m$ in ${\mathbb {Z}}^{n}$ is defined as the set of all points $p\in {\mathbb {Z}}^{n}$ such that

$\|p\|_{1}\leq m$

Counting Points

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

See Also

• Combinatorics

References

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