Multi-Index of Length $n$

Let $n\in\mathbb{N}$. A multi-index of length $n$ is an element

$$\alpha=(\alpha_1,\dots,\alpha_n)\in(\mathbb{N}\cup\{0\})^n.$$

That is, a multi-index of length $n$ is an ordered $n$-tuple of nonnegative integers. The zero multi-index of length $n$ is

$$0=(0,\dots,0).$$

For each index $i\in\{1,\dots,n\}$, the $i$th standard basis multi-index is

$$e_i=(0,\dots,0,1,0,\dots,0),$$

with $1$ in the $i$th position and $0$ elsewhere. If

$$\alpha=(\alpha_1,\dots,\alpha_n),\qquad \beta=(\beta_1,\dots,\beta_n)$$

are multi-indices of length $n$, we write

$$\beta\le \alpha$$

if and only if $\beta_i\le \alpha_i$ for every $i\in\{1,\dots,n\}$. In that case the difference

$$\alpha-\beta$$

is the multi-index of length $n$ defined componentwise by

$$\alpha-\beta=(\alpha_1-\beta_1,\dots,\alpha_n-\beta_n).$$

In particular, if $\alpha_i>0$, then $e_i\le \alpha$ and

$$\alpha-e_i=(\alpha_1,\dots,\alpha_{i-1},\alpha_i-1,\alpha_{i+1},\dots,\alpha_n).$$