Combinations without specified separations
Michael A. Allen

TL;DR
This paper studies subsets of natural numbers with restricted differences, establishing recursion relations and bijections with restricted-overlap tilings of boards using squares and comb-shaped tiles, to analyze combinatorial properties.
Contribution
It introduces a novel bijection between restricted subsets and overlap tilings with combs, providing new recursion relations for counting such subsets.
Findings
Derived recursion relations for the number of subsets with restricted differences.
Established a bijection between subsets and restricted-overlap tilings with combs.
Provided an intuitive method for analyzing combinatorial structures using tilings.
Abstract
We consider the restricted subsets of with being the largest member of the set of disallowed differences between subset elements. We obtain new results on various classes of problem involving such combinations lacking specified separations. In particular, we find recursion relations for the number of -subsets for any when . The results are obtained, in a quick and intuitive manner, as a consequence of a bijection we give between such subsets and the restricted-overlap tilings of an -board (a linear array of square cells of unit width) with squares ( tiles) and combs. A -comb is composed of sub-tiles known as teeth. The -th tooth in the comb has width and is separated from the -th tooth by a gap of width…
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Taxonomy
Topicsgraph theory and CDMA systems · Quasicrystal Structures and Properties · Cellular Automata and Applications
