Order and Factorial of a Multi-Index

Let $n\in\mathbb{N}$, and let

$$\alpha=(\alpha_1,\dots,\alpha_n)$$

be a multi-index of length $n$ in the sense of Multi-Index of Length $n$. The order of $\alpha$ is the nonnegative integer

$$|\alpha|=\alpha_1+\cdots+\alpha_n.$$

The factorial of $\alpha$ is the natural number

$$\alpha! = \alpha_1!\cdots \alpha_n!,$$

where each factorial on the right is the one from the factorial definition.