Background: Many textbooks give an example due to Peano of the function $f(x, y) = xy(x^2 - y^2)(x^2 + y^2)^{-1}$ that has mixed partial derivatives at $0$ that are not symmetric. One might wonder how arbitrary mixed partial derivatives can be at a point. For what choices of an appropriate number of real numbers is there a real-valued function that takes on those values for its partial derivatives at a point?
Notation:
Let $d\in \mathbb{N}$ and let $[d] = \{1,\ldots,d\}$. Let $\mathcal{I}$ be the set of all $n$-tuples with elements in $[d]$ (that is, $\mathcal{I} = \{(i_{1},\ldots, i_{n}): i_{1},\ldots, i_{n}\in [d]\}$). Let $h:\mathcal{I}\rightarrow \mathbb{R}$ . For which functions $h$ is there a function $f:\mathbb{R}^{d}\rightarrow \mathbb{R}$ so that$$\frac{\partial}{\partial x_{i_{n}}}\cdots \frac{\partial}{\partial x_{i_{1}}}f = h((i_{1},\ldots, i_{n}))$$at $x = 0\in \mathbb{R}^{d}$