Partition of a Set of Integers into Subsets with Prescribed Sums

Wang Yiju (yiju-66263.sina.com)

Abstract: A nonincreasing sequence of positive integers $\langle m_1,m_2,\cdots,m_k \rangle$ is said to be {\em $n$-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 $\sum\limits_{x\in S_i}x=m_i$ 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 $\sum\limits_{i=1}^km_i={n+1\choose 2}$ and $m_{k-1}\ge n$.

Keywords: Partition, Integer partition, Partite graph.

Category 1: Combinatorial Optimization (Graphs and Matroids )

Citation: 1, Inst. of Opersearchs Research, Qufu Normal university, Rizhao Shandong, 276800, China, Mar, 2003

Download:

Entry Submitted: 03/30/2003
Entry Accepted: 03/31/2003
Entry Last Modified: 04/02/2003

Modify/Update this entry