arXiv 1605.033222016-05-11math.CO

关于具有相同间隙序列的$4$ -集的整数拼接

On tiling the integers with $4$-sets of the same gap sequence

Ilkyoo Choi, Junehyuk Jung, Minki Kim

备注：: 12 pages, 4 figures

摘要：

将一个集合分割成相似的（如果不是相同的话）部分是组合学中的一个基本研究课题。在历史上，人们一直在考虑用各种方法划分整数的问题。给定一个整数集合$\{x_1, \ldots, x_n\}$，其中$x_1<\cdots<x_n$，设这个集合的{ \it间隔序列}是非减序列$d_1, \ldots, d_{n-1}$，其中$\{d_1, \ldots, d_{n-1}\}$等于作为多重集合的$\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$。本文解决了以下问题，这是明确地问Nakamigawa：一组整数可以划分成具有相同的间隙序列集？这个问题对于任何空位序列的长度最多为2的集合都是成立的。本文提供的证据表明，这个问题是真实的间隙序列的长度为3。也就是说，我们证明了给定正整数$p$和$q$，存在正整数$r_0$使得对所有$r\geq r_0$，整数集都能被划分成具有间隙序列$p, q$，$r$的$4$ -集.

英文摘要：

Partitioning a set into similar, if not, identical, parts is a fundamental research topic in combinatorics. The question of partitioning the integers in various ways has been considered throughout history. Given a set $\{x_1, \ldots, x_n\}$ of integers where $x_1<\cdots<x_n$, let the {\it gap sequence} of this set be the nondecreasing sequence $d_1, \ldots, d_{n-1}$ where $\{d_1, \ldots, d_{n-1}\}$ equals $\{x_{i+1}-x_i:i\in\{1,\ldots, n-1\}\}$ as a multiset. This paper addresses the following question, which was explicitly asked by Nakamigawa: can the set of integers be partitioned into sets with the same gap sequence? The question is known to be true for any set where the gap sequence has length at most two. This paper provides evidence that the question is true when the gap sequence has length three. Namely, we prove that given positive integers $p$ and $q$, there is a positive integer $r_0$ such that for all $r\geq r_0$, the set of integers can be partitioned into $4$-sets with gap sequence $p, q$, $r$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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