Is that function injective? Let $A \subset \mathbb{R}^n$ be defined by $(x_1, \dots ,x_n) \in A \Leftrightarrow x_1 < x_2 < \dots < x_n$.
Define $\phi \colon A \to \mathbb{R}^n$ by
$\begin{pmatrix}
x_1\\
\vdots\\
x_n
\end{pmatrix}
\mapsto
\begin{pmatrix}
\sum_j x_j\\
\sum_j (x_j)^2\\
\vdots\\
\sum_j (x_j)^n
\end{pmatrix}
$.
Can I now prove, that $\Phi$ is injective?
 A: This true up to a permutation of the $x_i$s, and is purely algebraic, because of the Newton-Girard relations between the elementary symmetric functions of $x_1,\dots, x_n$, which determine the set $\{x_1,\dots,x_n\}$,and Newton's sums (which are your sums of powers).
Some details:
Let $\begin{cases}
s_1=x_1+\dots+x_n\\ s_2=\!\!\!\sum\limits_{1\le i<j\le n}\!\!\! x_ix_j\\[-4ex]\vdots\\[-2ex]s_n=x_1\dots x_n\end{cases}\enspace$ be the symmetric functions in $n$ variables, and 
let $\begin{cases}
p_1=x_1+\dots+x_n\\ p_2= x_1^2+\dots +x_n^2\\[-4ex]\vdots\\[-2ex]p_k=x_1^k+\dots+ x_n^k\\\vdots
 \end{cases}\quad$ be the sums of powers of $n$ variables.
One has the following relations between the two kinds of sums:
\begin{align*}
s_1&=p_1,& 2s_2&= s_1p_1-p_2,\\
3s_3&=s_2p_1-s_1p_2+p_3,&4s_4&=s_3p_1-s_2p_2+s_1p_3-p_4,\\&\vdots&\vdots
\end{align*}
These relations show that, given the sums of powers $p_1, p_2,\dots, p_n$, the elementary symmetric functions $s_1, s_2, \dots, s_n$ are determined, hence the numbers $x_1, x_2, \dots, x_n$ are determined, and injectivity is proved.
