# A polynomial power-compositions determinant

Josep M. Brunat1

Departament de Matemàtica Aplicada II
Universitat Politècnica de Catalunya
C. Pau Gargallo, 5. E-08028 Barcelona, Catalonia, Spain
Josep.M.Brunat@upc.edu

Let and be positive integers. A -composition of is a -tuple of non-negative integers such that . Denote by the set of -compositions of . If and are -compositions of , we denote where to be consistent, it is assumed that . The power-compositions determinant is the determinant

The value of is given in [1]:

Recently, C. Krattenthaler in the complement [3] to its impressive Advanced Determinant Calculus[2], has given an equivalent formula for and has stated the following conjecture supported by computer experiments:

where is a variable and is short for . In this talk we prove this conjecture using a method that can be useful for other combinatorial determinants.

#### Footnotes

... Brunat1
joint work with Antonio Montes

2005-05-23