Usually, high order partial derivatives of a function $f:\mathbb{R}^n\to \mathbb{R}$ can be described by a multiindex $\mathbf{k}=(k_1,\dots, k_n)\in \mathbb{N}_0^n$, see for instance http://en.wikipedia.org/wiki/Multi-index_notation. The quantity $k_i$ describes the number of partial derivatives in the $i$-th direction. The reason why this notation captures all cases of high order partial derivatives is that the order of differentiation is irrelevant, so that the number of derivatives in each coordinate completely describes a partial differential operator.

Suppose, we would like to describe high order partial covariant derivatives of a function $f:\mathbb{R}^n\to M$ for a Riemannian manifold $M$, for instance (n=2) $$\frac{D}{dx^1}\frac{D}{dx^2}\frac{d}{dx^1}f.$$

Since, in general, the order of differentiation is relevant, one cannot simply describe this operation by a multiindex $\mathbf{k}=(k_1,\dots, k_n)\in \mathbb{N}_0^n$.

For example, if we use the multiindex notation for the covariant derivative above, we would get the multiindex $(2,1)$, which would equally correspond to the operator $$\frac{D}{dx^2}\frac{D}{dx^1}\frac{d}{dx^1}f$$ which is different from the original covariant derivative operator.

My question: Does there exist a unified notation for high order partial covariant derivatives for functions $f:\mathbb{R}^d\to M$?