  Following the ideas of L. Carlitz we introduce a generalization of the
Bernoulli and Eulerian polynomials of higher order to vectorial index and
argument. These polynomials are used for computation of the vector partition
function $W({\bf s},{\bf D})$, i.e., a number of integer solutions to a linear
system ${\bf x} \ge 0, {\bf D x} = {\bf s}$. It is shown that $W({\bf s},{\bf
D})$ can be expressed through the vector Bernoulli polynomials of higher order.
