Partition of a set of integers into subsets, with prescribed sums

A nonincreasing sequence of positive integers < m(1), m(2),(...), m(k)> is said to be n-realizable if the set I-n = {1, 2,(...), n} can be partitioned into k mutually disjoint subsets S-1, S-2,(...), S-k such that Sigma(x is an element of Si) x = m(i) for each 1. <= i <= k. In this paper, we will prove. that a nonincreasing sequence of positive integers < m(1), m(2),(...),m(k)> is n-realizable under the. conditions that Sigma(i=1)(k) m(i) = ((n+1)(2)) and m(k-1) >= n.