Partition of a set of integers into subsets with prescribed sums

A nonincreasing sequence of positive integers (m₁, 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 S₁, S₂, ⋯, S_k such that ∑_x∈si x = m_i for each 1 ≤ i ≤ k. In this paper, we will prove that a nonincreasing sequence of positive integers (m ₁, m₂, ⋯, m_k) is n-realizable under the conditions that ∑_i=1 ^k m_i = (n+1/2) and m_k-1 ≥ n.