is a grungy technical tool which is used when writing down explicit formulas for the basic theorems of multivariable calculus
al Euclidean space
A multi-index of dimension n and weight k is an element α = (α1, ..., αn) ∈ Nn, that is an n-tuple of nonnegative integers, which adds up to k:
|α| = α1 + ... + αn = k.
The number of different multi-indices of dimension n and weight k is C(k + n - 1, k) = C(k + n - 1, n - 1), which you can compute using Stars and Bars.
We then extend the usual notation for partial derivatives of functions f: Rn → R to handle multi-indices. Let ∂jf abbreviate the partial derivative ∂f/∂xj. Then we write
∂αf for (∂1)α1 ... (∂n)αn f, and
α! for α1! ... αn!
The multi-index notation is similar in purpose to physicists' explicit tensor calculus (which snooty pure mathematicians have called the "debauch of indices") in that it attempts to address the necessity for doing concrete calculations while maintaining some degree of conciseness and understandability. In partial differential equations often the multi-index formulation of a theorem is preferred to the "beautiful" coordinate-free one for this reason. See Leibniz rule and Taylor's formula for examples.