Current Search: Meyerowitz, Aaron (x)


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 3subsets 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 3subsets 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)


Title

Bijections for partition identities.

Creator

Lai, JinMei 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 sieveequivalent 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 sieveequivalent 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)