I am interested in the following question:
Let $,kn$ be a positive integeres.
Assume $\sum_{i=1}^{k} L_i=\sum_{i=1}^{k+1} \tilde L_i=n$, where $L_i,\tilde L_i$ are positive integers.
Is it true that $\sum_{i=1}^{k} L_i^2 > \sum_{i=1}^{k+1} (\tilde L_i)^2$?
(i.e does the sum of squares necessarily get smaller when the number of terms increase?)
The motivation actually comes from this question of mine. In an answer there, it is proved that the dimension of a certain subspace of matrices depends on a sum of squares of this form.
