# A polynomial power-compositions determinant

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

- ... Brunat
^{1}
- joint work with Antonio Montes

2005-05-23