The Minimal Sum of Squares Over Partitions with a Nonnegative Rank

Abstract

Motivated by a question of Defant and Propp (Electron J Combin 27:Article P3.51, 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.

Appendix

Denote $z_n=\left\lfloor x_0^{(n)}\right\rfloor$ and $X_i=\frac{1}{(z_n-1)^i}$, for $i=1,2,3$. We have

$$\begin{aligned} \frac{p_n^3}{27}+\frac{q_n^2}{4}&=\boxed {-\frac{n^{6}}{27}X_{3}} +\frac{2n^{5}z_{n}}{9}X_{3}-\frac{2n^{5}}{9}X_{2}\\&\quad -\frac{5n^{4}z_{n}^{2}}{9}X_{3} \boxed {-\frac{n^{4}z_{n}^{2}}{9}X_{2}}+\frac{31n^{4}z_{n}}{36}X_{2} \boxed {+\frac{n^{4}}{4}}\\&\quad +\frac{5n^{4}}{18}X_{2}+\frac{n^{4}}{18}X_{1} +\frac{20n^{3}z_{n}^{3}}{27}X_{3}\\&\quad +\frac{4n^{3}z_{n}^{3}}{9}X_{2} -\frac{11n^{3}z_{n}^{2}}{9}X_{2}-\frac{4n^{3}z_{n}^{2}}{9}X_{1}\\&\quad -\frac{10n^{3}z_{n}}{9}X_{2}-\frac{2n^{3}z_{n}}{9}X_{1} -\frac{8n^{3}}{27}+\frac{n^{3}}{9}X_{1}\\&\quad -\frac{5n^{2}z_{n}^{4}}{9}X_{3} -\frac{2n^{2}z_{n}^{4}}{3}X_{2}\boxed {-\frac{n^{2}z_{n}^{4}}{9}X_{1}} \\&\quad +\frac{13n^{2}z_{n}^{3}}{18}X_{2}+\frac{5n^{2}z_{n}^{3}}{6}X_{1} +\frac{n^{2}z_{n}^{2}}{18}+\frac{5n^{2}z_{n}^{2}}{3}X_{2}\\&\quad +\frac{119n^{2}z_{n}^{2}}{144}X_{1}+\frac{n^{2}z_{n}}{72} -\frac{n^{2}z_{n}}{12}X_{1}\\&\quad -\frac{n^{2}}{72}-\frac{19n^{2}}{144}X_{1} +\frac{2nz_{n}^{5}}{9}X_{3}+\frac{4nz_{n}^{5}}{9}X_{2}\\&\quad +\frac{2nz_{n}^{5}}{9}X_{1}-\frac{2nz_{n}^{4}}{9} -\frac{nz_{n}^{4}}{9}X_{2}-\frac{nz_{n}^{4}}{3}X_{1} \\&\quad -\frac{nz_{n}^{3}}{9}-\frac{10nz_{n}^{3}}{9}X_{2} -\frac{29nz_{n}^{3}}{24}X_{1}+\frac{7nz_{n}^{2}}{72}\\&\quad -\frac{nz_{n}^{2}}{6}X_{1}+\frac{nz_{n}}{36} +\frac{19nz_{n}}{72}X_{1}-\frac{n}{72}\\&\quad \boxed {-\frac{z_{n}^{6}}{27}}-\frac{z_{n}^{6}}{27}X_{3} -\frac{z_{n}^{6}}{9}X_{2}-\frac{z_{n}^{6}}{9}X_{1} -\frac{z_{n}^{5}}{36}\\&\quad -\frac{z_{n}^{5}}{36}X_{2} -\frac{z_{n}^{5}}{18}X_{1}+\frac{13z_{n}^{4}}{48} +\frac{5z_{n}^{4}}{18}X_{2}\\&\quad +\frac{79z_{n}^{4}}{144}X_{1}+\frac{239z_{n}^{3}}{1728} +\frac{5z_{n}^{3}}{36}X_{1}-\frac{11z_{n}^{2}}{96} -\frac{19z_{n}^{2}}{144}X_{1}-\frac{19z_{n}}{576}+\frac{7}{432}. \end{aligned}$$

Recall that $\lim _{n\rightarrow \infty } \frac{z_n}{n^{2/3}} = 2^{-1/3}$ (this was stated in the proof of Theorem 2.3). Thus, the expansion of $\frac{p_n^3}{27}+\frac{q_n^2}{4}$ contains terms of order $n^4$ (the boxed terms). Nevertheless, the overall order is $<n^4$, as the following calculation shows:

$$\begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{n^4}\left( -\frac{n^{6}}{27}X_{3} -\frac{n^{4}z_{n}^{2}}{9}X_{2}+\frac{n^{4}}{4} -\frac{n^{2}z_{n}^{4}}{9}X_{1}-\frac{z_{n}^{6}}{27}\right) \\&\quad =\lim _{n\rightarrow \infty }\frac{1}{108}\left( -4\left( \frac{n^{2/3}}{z_n} \right) ^3\frac{z_n^3}{(z_n-1)^3}-12\frac{z_n^2}{(z_n-1)^2}\right. \\&\qquad \left. +27 -12\left( \frac{z_{n}}{n^{2/3}}\right) ^3\frac{z_n}{z_n-1} -4\left( \frac{z_{n}}{n^{2/3}}\right) ^6\right) \\&\quad =\lim _{n\rightarrow \infty }\frac{1}{108}\left( -4\cdot 2-12+27-12\cdot 2^{-1} -4\cdot 2^{-2}\right) =0. \end{aligned}$$