On tiling the integers with 4-sets of the same gap sequence |
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Authors: | Ilkyoo Choi Junehyuk Jung Minki Kim |
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Institution: | 1. Department of Mathematics, Hankuk University of Foreign Studies, Yongin-si, Gyeonggi-do, Republic of Korea;2. Department of Mathematics, Texas A&M University, TX, United States;3. Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea |
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Abstract: | 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 of integers where , let the gap sequence of this set be the unordered 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 and , there is a positive integer such that for all , the set of integers can be partitioned into 4-sets with gap sequence , . |
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Keywords: | Integer partitions Gap sequence Tilings |
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