On Sunday, September 8, 2013 5:16:46 AM UTC-7, 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' ??
<quote> The systematic study of structures in words was started by Axel Thue in 1906. In 1961, Paul Erdös raised the question whether abelian squares can be avoided in infinitely long words. An abelian square means a non-empty word uv, where u and v are permutations of each other. For example, abc acb is an abelian square. A word is called abelian square-free if it does not contain any abelian square as a factor. For example, the word abacaba is abelian square-free, while ab cabdc bcacd ac is not. </quote>
From "Abelian Square-Free Partial Words" by Francine Blanchet-Sadri, Jane I. Kim, Robert Merca?, William Severa, Sean Simmons <quote> Words or strings belong to the very basic objects in theoretical computer science. The systematic study of word structures (combinatorics on words) was started by a Norwegian mathematician Axel Thue at the beginning of the last century. One of the remarkable discoveries made by Thue is that the consecutive repetitions of non-empty factors (squares) can be avoided in infinite words overa three-letter alphabet. Recall that an infinite word w over an alphabet is said to be k-free if there exists no word x such that xk is a factor of w. For simplicity, a word that is 2-free is said to be square-free. Erdos raised the question whether abelian squares can be avoided in infinitely long words, i.e., whether there exist infinite abelian square-free words over a given alphabet. An abelian square is a non-empty word uv, where u and v are permutations of each other. For example, abcacb is an abelian square. A word is called abelian square-free, if it does not contain any abelian square as a factor. For example, the word abacaba is abelian square-free, while abcdadcada is not (it contains the subword cdadca). </quote>