I asked [this question on math.SE](https://math.stackexchange.com/questions/1405826/integer-points-in-simplex) before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..


Let $q \in \mathbb{R}\_+$ and $0 < w\_1 \leq w\_2 \leq \ldots w\_d \in \mathbb{R}\_+$. Define
$$A\_w(d,q):=\left\{{\bf k} \in \mathbb{N}\_0^d: \sum\_{j=1}^d w\_j k\_j \leq q\right\}$$
denote the number of non-negative integer points in the $\ell\_1$-ellipse with semi-axes of length $\frac{q}{w\_j}$ (which is a simplex).
Similarly, 
$$B\_w(d,q):=\left\{{\bf k} \in \mathbb{N}\_+^d: \sum\_{j=1}^d w\_j k\_j \leq q\right\}$$
denotes the number of positive integer points in this simplex.


My question is: Is there a relationship between $A\_w(d,q)$ and $B\_w(d,q)$ in the sense that if one could obtain an exact formula for $A\_w(d,q)$ there also would be an exact formula for $B\_w(d,q)$? For example something like
$$B\_w(d,q) = A\_\tilde{w}(d,\tilde{q}) .$$


Thanks!



Relation between number of non-negative and positive integers points in simplices
I found a solution to the problem in this paper -- formulas (1.2) and (1.3)


<http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf>


