For sums of N increasing real sequences, the paper determines the maximal multiplicity of the k-th sum exactly for N=2 and proves a lower bound for N=3.
Cameron, Combinatorics: Topics, Techniques, Algorithms
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On maximal multiplicities for Hamiltonians with separable variables
For sums of N increasing real sequences, the paper determines the maximal multiplicity of the k-th sum exactly for N=2 and proves a lower bound for N=3.