# Lewko's blog

## Lattice points in l^{1} balls

Posted in math.CO, math.NT by Mark Lewko on July 31, 2009

For positive integers ${n}$ and ${m}$, let ${L(n,m)}$ denote the number of points ${x = (x_1, \ldots, x_n) \in \mathbb{Z}^n}$ such that ${\sum_{i=1}^n |x_i| \leq m}$.  In other words, ${L(n,m)}$ is the number of lattice points in an $n$-dimensional $l^1$-ball of radius $m$Bump, Choi, Kurlberg and Vaaler have observed that the function ${L(n,m)}$ enjoys the following symmetry

$\displaystyle {L(n,m) = L(m,n)}$

This is to say that the $n$-dimensional $l^1$-ball of radius $m$ contains exactly the same number of lattice points as the $m$-dimensional $l^1$-ball of radius $n$.  The original proof follows from establishing the generating function identity

$\displaystyle \sum_{n=1}^{\infty}\sum_{m=1}^{\infty}L(m,n)x^m y^n = (1-x-y-xy)^{-1}$

and noting that it is symmetric in $x$ and $y$.  Some time ago Jeff Vaaler mentioned to me that it would be nice to find a bijective proof of this fact.  In this post I will give a simple bijective proof that Allison Bishop and I found.

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