  Motivated by a question of Defant and Propp (2020) regarding the connection
between the degrees of noninvertibility of functions and those of their
iterates, we address the combinatorial optimization problem of minimizing the
sum of squares over partitions of $n$ with a nonnegative rank. Denoting the
sequence of the minima by $(m_n)_{n\in\mathbb{N}}$, we prove that
$m_n=\Theta\left(n^{4/3}\right)$. Consequently, we improve by a factor of $2$
the lower bound provided by Defant and Propp for iterates of order two.
