发表机构
Warsaw University of Technology; Silesian University of Technology; AGH University of Krakow（华沙理工大学; 西里西亚理工大学; 克拉科夫AGH科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究有限与无限barrycade结构，证明存在无穷多个规模具有线性高度的barrycade，并提出三种构造方法及若干猜想。
AI 中文摘要
高度为 $h$ 的 $n$-barrycade 是 $[n]$ 的 $h$ 个排列的集合，使得所有前缀和均不相同。这一概念由 Richard Guy 于 2020 年提出，同时提出了核心问题：确定对于哪些 $n$ 值存在无断裂的 $n$-barrycade，即其中从 $1$ 到 $\ rac{n(n+1)}{2}-1$ 的每个可能前缀和都被覆盖。此类 barrycade 的高度应为 $\ rac{n+2}{2}$。我们证明了线性高度的 barrycade 存在于无穷多个规模中：存在常数 $c>0$，使得对于无穷多个正整数 $n$，存在高度至少为 $cn$ 的 $n$-barrycade。我们还探讨了该问题的无限变体，并提出了三种构造——贪心（greedy）、蚱蜢（grasshopper）和精确蚱蜢（precise grasshopper）——它们给出了越来越令人满意的结果。在此过程中，我们遇到了新的整数序列，并针对省略元素、缺失部分和以及无限 barrycade 的单词表示提出了猜想，这些猜想得到了计算数据的坚实支持。
英文摘要
An $n$-barrycade of height $h$ is a set of $h$ permutations of $[n]$ such that all the prefix sums are different. This notion was described in 2020 by Richard Guy, together with the central problem to determine for which values of $n$ there exists a break-free $n$-barrycade, that is, one in which every possible prefix sum from $1$ to $\frac{n(n+1)}{2}-1$ is covered. The height of such barrycade would be $\frac{n+2}{2}$. We prove that barrycades of linear height exist for infinitely many sizes: there is a constant $c>0$ such that for infinitely many positive integers $n$ there exists an $n$-barrycade of height at least $cn$. We also explore an infinite variant of the problem and propose three constructions -- greedy, grasshopper and precise grasshopper -- that give increasingly more satisfying results. Along the way we encounter new integer sequences and formulate conjectures concerning omitted elements, missing partial sums, and word representations of infinite barrycades, which are firmly supported by computational data.
Comments18 pages, 6 figures
