PARTITION OF A SET OF INTEGERS INTO SUBSETS WITH PRESCRIBED SUMS

Abstract

A nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is said to be $n$-realizable if the set $I_n = \{1,2,\cdots,n\}$ can be partitioned into $k$ mutually disjoint subsets $S_1,S_2,\cdots,S_k$ such that $\sum_{x \in S_i} x = m_i$ for each $1 \le i \le k$. In this paper, we will prove that a nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is $n$-realizable under the conditions that $\sum_{i=1}^k m_i = \binom{n+1}{2}$ and $m_{k-1} \ge n$.