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Tiling Z with Triples Using Signed Permutation Matrices
Title
Tiling Z with Triples Using Signed Permutation Matrices.
Creator
Cattell, Liam J., Meyerowitz, Aaron, Florida Atlantic University
Abstract/Description
The topic of this paper is tiling the integers with triples, or more precisely to write Z as a disjoint union of translates of a given set of 3-subsets composed of basic shapes called prototiles. We fix the set of proto tiles P = { { 0, a, a+ v} , { U. b, a+ b}} and define an algorithm which returns a sequence of translates of P when given an initial subset of Z representing integers that are already tiled. This algorithm is then adapted to describe all possible tilings with triples from P...
Show moreThe topic of this paper is tiling the integers with triples, or more precisely to write Z as a disjoint union of translates of a given set of 3-subsets composed of basic shapes called prototiles. We fix the set of proto tiles P = { { 0, a, a+ v} , { U. b, a+ b}} and define an algorithm which returns a sequence of translates of P when given an initial subset of Z representing integers that are already tiled. This algorithm is then adapted to describe all possible tilings with triples from P using the action of certain signed permutation matrices on a subset of za+b , uamdy the 2" Yectors with all entries ±1. Given b > 2a, we research properties of the digraph of all possible tiling states and some related digraphs.
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Date Issued
2007
PURL
http://purl.flvc.org/fau/fd/FA00000732
Subject Headings
Tiling (Mathematics), Sequences (Mathematics), Permutation groups
Format
Document (PDF)
Bijections for partition identities
Title
Bijections for partition identities.
Creator
Lai, Jin-Mei Jeng, Florida Atlantic University, Meyerowitz, Aaron, Charles E. Schmidt College of Science, Department of Mathematical Sciences
Abstract/Description
This paper surveys work of the last few years on construction of bijections for partition identities. We use the more general setting of sieve--equivalent families. Suppose A1' ... ,An are subsets of a finite set A and B1' ... ,Bn are subsets of a finite set B. Define AS=∩(i∈S) Ai and BS = ∩ (i∈S) Bi for all S⊆N={1,...,n}. Given explicit bijections fS: AS->BS for each S⊆N, A-∪Ai has the same size as B-∪Bi. Several authors have given algorithms for producing an explicit bijection between these...
Show moreThis paper surveys work of the last few years on construction of bijections for partition identities. We use the more general setting of sieve--equivalent families. Suppose A1' ... ,An are subsets of a finite set A and B1' ... ,Bn are subsets of a finite set B. Define AS=∩(i∈S) Ai and BS = ∩ (i∈S) Bi for all S⊆N={1,...,n}. Given explicit bijections fS: AS->BS for each S⊆N, A-∪Ai has the same size as B-∪Bi. Several authors have given algorithms for producing an explicit bijection between these two sets. In certain important cases they give the same result. We discuss and compare algorithms, use Graph Theory to illustrate them, and provide PAS CAL programs for them.
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Date Issued
1992
PURL
http://purl.flvc.org/fau/fd/FADT14826
Subject Headings
Algorithms, Partitions (Mathematics), Sieves (Mathematics)
Format
Document (PDF)