Let $C(n,p)$ be the set of $p$-compositions of an integer $n$, i.e., the setof $p$-tuples $\bm{\alpha}=(\alpha_1,...,\alpha_p)$ of nonnegative integerssuch that $\alpha_1+...+\alpha_p=n$, and $\mathbf{x}=(x_1,...,x_p)$ a vector ofindeterminates. For $\bm{\alpha}$ and ${\bm{\beta}}$ two $p$-compositions of$n$, define $(\mathbf{x}+\bm{\alpha})^{\bm{\beta}} =(x_1+\alpha_1)^{\beta_1}... x_p+\alpha_p)^{\beta_p}$. In this paper we prove anexplicit formula for the determinant $\det_{\bm{\alpha},{\bm{\beta}}\inC(n,p)}((\mathbf{x}+\bm{\alpha})^{\bm{\beta}})$. In the case $x_1=...=x_p$ theformula gives a proof of a conjecture by C.~Krattenthaler.
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