Partition of a set of integers into subsets with prescribed sums

Abstract

A nonincreasing sequence of positive integers (m1, m 2, ⋯, mk) is said to be n-realizable if the set In = {1, 2, ⋯, n} can be partitioned into k mutually disjoint subsets S1, S2, ⋯, Sk such that ∑x∈si x = mi for each 1 ≤ i ≤ k. In this paper, we will prove that a nonincreasing sequence of positive integers (m 1, m2, ⋯, mk) is n-realizable under the conditions that ∑i=1k mi = (n+1/2) and mk-1 ≥ n.