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I asked this question on math.SE 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!

I asked this question on math.SE 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!

I asked this question on math.SE 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!

I asked this question on math.SE 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!

I asked this question on math.SE before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..

Let $$A_w(d,q):=\left\{{\bf k} \in \mathbb{N}_0^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ 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.

We can assume that $0 < w_1 \leq w_2 \ldots \leq w_d$ and $q \geq w_d$.

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!

I asked this question on math.SE 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!