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 $\mathbf{\alpha }=({\alpha }_1,...,{\alpha }_p)$ of nonnegative integers such that ${\alpha }_1+...+{\alpha }_p=n$, and $\mathbf{x}=(x_1,...,x_p)$ a vector of indeterminates. For $\mathbf{\alpha }$ and $\mathbf{\beta }$ two $p$-compositions of $n$, define $(\mathbf{x}+\mathbf{\alpha }{)}^{\mathbf{\beta }}=(x_1+{\alpha }_1{)}^{{\beta }_1}...x_p+{\alpha }_p{)}^{{\beta }_p}$. In this paper we prove an explicit formula for the determinant ${\det }_{\mathbf{\alpha },\mathbf{\beta }\in C(n,p)}((\mathbf{x}+\mathbf{\alpha }{)}^{\mathbf{\beta }})$. In the case $x_1=...=x_p$ the formula gives a proof of a conjecture by C.~Krattenthaler.

11 pages, see also http://www-ma2.upc.edu/~montes/