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 $\bmα=(α_1,...,α_p)$ of nonnegative integers such that $α_1+...+α_p=n$, and $\mathbf{x}=(x_1,...,x_p)$ a vector of indeterminates. For $\bmα$ and ${\bmβ}$ two $p$-compositions of $n$, define $(\mathbf{x}+\bmα)^{\bmβ} = (x_1+α_1)^{β_1}... x_p+α_p)^{β_p}$. In this paper we prove an explicit formula for the determinant $\det_{\bmα,{\bmβ}\in C(n,p)}((\mathbf{x}+\bmα)^{\bmβ})$. In the case $x_1=...=x_p$ the formula gives a proof of a conjecture by C.~Krattenthaler.