Figurate numbers and sums of powers of integers

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$F_{n}^{p-\ell}$isthe$n$-thhyper-tetrahedronnumberofdimension$p-\ell$and$c_{p,\ell}$denotesthenumberof$(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0\le x_1,x_2,\dots ,x_p\le n-1$. 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 $c_{p,\mathrm{\ell }}=(p-\mathrm{\ell })!S(p,p-\mathrm{\ell })$, where $S(p,p-\mathrm{\ell })$ 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.