The Minimal Sum of Squares Over Partitions with a Nonnegative Rank

IF 0.6 4区数学 Q4 MATHEMATICS, APPLIED

Annals of Combinatorics Pub Date : 2022-12-03 DOI:10.1007/s00026-022-00625-z

Sela Fried

引用次数: 0

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.

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非负秩分区上的最小平方和

受Defant和Propp（Electron J Combin 27:文章P3.51/2020）关于函数的不可逆度与其迭代函数的不可可逆度之间的联系的问题的启发，我们解决了在具有非负秩的n的分区上最小化平方和的组合优化问题。用\（（m_n）_｛n\in｛\mathbb｛n｝｝）｝\表示极小值的序列，我们证明了\（m_n=\Theta\left（n^｛4/3｝\right）\）。因此，我们将Defant和Propp为二阶迭代提供的下界提高了2倍。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Annals of Combinatorics 数学-应用数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Annals of Combinatorics publishes outstanding contributions to combinatorics with a particular focus on algebraic and analytic combinatorics, as well as the areas of graph and matroid theory. Special regard will be given to new developments and topics of current interest to the community represented by our editorial board. The scope of Annals of Combinatorics is covered by the following three tracks: Algebraic Combinatorics: Enumerative combinatorics, symmetric functions, Schubert calculus / Combinatorial Hopf algebras, cluster algebras, Lie algebras, root systems, Coxeter groups / Discrete geometry, tropical geometry / Discrete dynamical systems / Posets and lattices Analytic and Algorithmic Combinatorics: Asymptotic analysis of counting sequences / Bijective combinatorics / Univariate and multivariable singularity analysis / Combinatorics and differential equations / Resolution of hard combinatorial problems by making essential use of computers / Advanced methods for evaluating counting sequences or combinatorial constants / Complexity and decidability aspects of combinatorial sequences / Combinatorial aspects of the analysis of algorithms Graphs and Matroids: Structural graph theory, graph minors, graph sparsity, decompositions and colorings / Planar graphs and topological graph theory, geometric representations of graphs / Directed graphs, posets / Metric graph theory / Spectral and algebraic graph theory / Random graphs, extremal graph theory / Matroids, oriented matroids, matroid minors / Algorithmic approaches