The minimal sum of squares over partitions with a nonnegative rank

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.