```Date: Sep 8, 2013 12:03 PM
Author: fom
Subject: Re: abelian square

On 9/8/2013 7:16 AM, Dieter von Holten wrote:> hi folks,>> i found 'abelian squares' in the context of patterns of n symbols of length 2n like 'abcd.cdba' where 'cdba' is a permutation of the first half 'abcd'.> however, in those texts it is not explained (maybe just not clear enough for me) what is 'abelian' and where is the 'square' ??>> any hints ?>> thanks> dvh>Looking at the definition inhttp://arxiv.org/pdf/0807.5028v1.pdfbothabcd.cdabcdab.abcdsatisfy the constraint that the secondstring is a permutation of the first.Since this is true of all finite strings,one would expect that "abelian" referssimply to the fact that the two stringscommute over the separator, '.', relativeto the definition.As for being a "square", that would seemto be a trickier guess.  If you look atsection 2 of the paper,http://poncelet.math.nthu.edu.tw/disk5/js/computer-science/on_abelian_squares_and_substitut.pdfyou will find that the Parikh vector mentionedin the first paper is the basis for anequivalence class as defined in thesecond paper.Relative to this equivalence class, onehas[x] = [x'][x][x'] = [x]^2 = [x']^2So, this would be my guess concerning thesense of "square" in the name.
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