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I find an interesting matrix which seems to be always invertible. But I have no idea how to prove it! So I write down here for some ideas. Here is the problem:

Let us take $n\in \mathbb{N}^*$ bins and $d\in \mathbb{N}^*$ balls. Denote the set $B = \{\alpha^1, \ldots, \alpha^m\}$ to be all possible choices for putting $d$ balls into $n$ bins (empty bin is possible), such as

$$\alpha^1 = (d,0,\ldots, 0), ~ \alpha^2 = (0,d,\ldots, 0), \ldots$$

Let us define the matrix $V$ as:

$$V = \begin{bmatrix}

(\alpha^1)^{\alpha^1} & \cdots & (\alpha^1)^{\alpha^m}\\

(\alpha^2)^{\alpha^1} & \cdots & (\alpha^2)^{\alpha^m}\\

\vdots & \vdots & \vdots\\

(\alpha^m)^{\alpha^1} & \cdots & (\alpha^m)^{\alpha^m}

\end{bmatrix}$$

where the notation $(\alpha^i)^{\alpha^j} = \displaystyle\prod_{k=1}^{n}(\alpha^i_k)^{\alpha^j_k}$, under the assumption that $0^0=1$.

**Question:**"

**is the matrix $V$ invertible?**"

I have tested several examples, and it seems that $V$ is always invertible, but I have no idea how to prove it or find a counterexample.

Here are two facts I understood:

- all diagonal elements are strictly positives, so the trace of $V$ is strictly positive.
- the matrix $V$ is not symmetric.