  Recently, Marko and Litvinov (ML) conjectured that, for all positive integers
$n$ and $p$, the $p$-th power of $n$ admits the representation $n^p =
\sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where
$F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$
and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by
cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In
this paper we show that the ML conjecture is true for every natural number $p$.
Our proof relies on the fact that the validity of the ML conjecture necessarily
implies that $c_{p,\ell} = (p-\ell)! S(p, p-\ell)$, where $S(p,p-\ell)$ are the
Stirling numbers of the second kind. Furthermore, we provide a number of
equivalent formulas expressing the sum of powers $\sum_{i=1}^{n} i^p$ as a
linear combination of figurate numbers.
