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Partition of a set of integers into subsets with prescribed sums. (English) Zbl 1093.11016

A nonincreasing sequence of positive integers \(\langle m_1, m_2, \ldots, m_k \rangle\) is said to be \(n\)-realizable if the set \(I_n = \{ 1, 2, \ldots, n\}\) can be partitioned into \(k\) mutually disjoint sunsets \(S_1, S_2, \ldots, S_k\) such that \(\sum_{x \in S_i} x = m_i\) for each \(i\), \(1 \leq i \leq k\).
The authors show that if a nonincreasing sequence of positive integers \(\langle m_1, m_2, \ldots, m_k \rangle\) satisfies \(\sum_{i=1}^k m_i = \) \(n \choose 2\) and \(m_{k-1} \geq n\), then this sequence is \(n\)-realizable. Moreover, they illustrate that this result cannot be improved by replacing the latter condition with \(m_{k-2} \geq n\).

MSC:

11B75 Other combinatorial number theory
90C35 Programming involving graphs or networks
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05A18 Partitions of sets
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