A polynomial generalization of the power-compositions determinant

Abstract

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.