Abstract
Let C(n, p) be the set of p-compositions of an integer n, i.e., the set of p-tuples α=(α1,… ,α
p
) of nonnegative integers such that α1+··· +α
p
=n, and x=(x1
,…,xp
) a vector of indeterminates. For α and β the two p-compositions of n, define
. In this article we prove an explicit formula for the determinant
. In the case x1=··· =xp
the formula gives a proof of a conjecture by C. Krattenthaler.
Acknowledgements
The authors would like to thank the referee very much for the valuable suggestions, corrections and comments, which resulted in a great improvement of the original manuscript. This work was partially supported by the Ministerio de Ciencia y Tecnología under projects BFM2003-00368, and MTM2004-01728 and Ministerio de Ciencia y Tecnología and by the Generalitat de Catalunya under project 2001 SGR 00224.