Decision problems for some classes of integer partitions and number multisets

We show that the class of integer partitions that cannot be represented as convex combinations of two partitions of the same number coincides with the class of knapsack partitions and the class of Sidon multisets, which includes sum-free sets and standard Sidon sets. The decision problem for knapsack partitions is proved to be co-$NP$-complete, and, therefore, it cannot be solved in polynomial time unless $P=NP$. Bibl. 13 titles.