Let $C(n,p)$ be the set of $p$-compositions of an integer $n$, i.e., the set of $p$-tuples $\bm{ \alpha}=(\alpha_1,...,\alpha_p)$ of nonnegative integers such that $\alpha_1+...+\alpha_p=n$, and $\mathbf{ x}=(x_1,...,x_p)$ a vector of indeterminates. 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 an explicit formula for the determinant $\det_{ \bm{ \alpha},{ \bm{ \beta}}\in C(n,p)}((\mathbf{ x}+\bm{ \alpha})^{ \bm{ \beta}})$. In the case $x_1=...=x_p$ the formula gives a proof of a conjecture by C.~Krattenthaler.
Categories: math.CO