On the Sylvester program and Cayley algorithm for vector partition reduction

A vector partition problem asks for a number of nonnegative integer solutions to a system of several linear Diophantine equations with integer nonnegative coefficients. J.J. Sylvester put forward an idea of reduction of vector partition to a sum of scalar partitions. In the simplest case of two equations with positive coefficients A. Cayley performed a reduction of the corresponding double partition to a sum of scalar partitions using an algorithm subject to a set of conditions on the coefficients. We suggested a modification of the original Cayley algorithm for the cases when these conditions are not satisfied. This result is extended to arbitrary number of the Diophantine equations to accomplish the Sylvester program of the vector partition reduction to a combination of scalar partitions.