The minimal sum of squares over partitions with a nonnegative rank

TL;DR

This paper investigates a combinatorial optimization problem involving partitions with nonnegative rank, establishing the asymptotic behavior of the minimal sum of squares and improving existing bounds for iterates of order two.

Contribution

It proves that the minimal sum of squares over such partitions grows asymptotically as n^{4/3}, refining previous bounds for function iterates.

Findings

The minimal sum of squares scales as Θ(n^{4/3}).

The result improves the lower bound for iterates of order two.

Provides a connection between partition properties and function noninvertibility.

Abstract

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.