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One-dimensional Tilings Using Tiles with Two Gap Lengths

Research paper by Tomoki Nakamigawa

Indexed on: 01 Mar '05Published on: 01 Mar '05Published in: Graphs and Combinatorics



Abstract

Let n,p,k,q,l be positive integers with n=k+l+1. Let x1,x2, . . . ,xn be a sequence of positive integers with x1<x2<···<xn. A set {x1,x2, . . . ,xn} is called a set of type (p,k;q,l) if the set of differences {x2−x1,x3−x2, . . . ,xn−xn−1} equals {p, . . . ,p,q, . . . ,q} as a multiset, where p and q appear k and l times, respectively. Among other results, it is shown that for any p,k,q, there exists a finite interval I in the set of integers such that I is partitioned into sets of type (p,k;q,1).