Figurate numbers and sums of powers of integers

18 Feb 2020 Cereceda José L.

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 of the $p$-dimensional simplex $x_{\sigma_1} \geq x_{\sigma_2} \geq \cdots \geq x_{\sigma_p}$ (where $\sigma$ is a permutation of $\{ 1, 2, \ldots, p \}$) 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 $p$... (read more)

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