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

be a multi-index of length $n$ in the sense of \ref{def:multi-index-length-n-2026a}. The order of $\alpha$ is the nonnegative integer

The factorial of $\alpha$ is the natural number

where each factorial on the right is the one from \reftext{def:factorial-natural-number-2026a}{the factorial definition}.