A sharpening of the Parikh mapping
TLDR
A sharpening of the Parikh map- ping is introduced and an interesting in- terconnection between mirror images of words and inverses of matrices is established.Abstract:
In this paper we introduce a sharpening of the Parikh map- ping and investigate its basic properties. The new mapping is based on square matrices of a certain form. The classical Parikh vector appears in such a matrix as the second diagonal. However, the matrix prod- uct gives more information about a word than the Parikh vector. We characterize the matrix products and establish also an interesting in- terconnection between mirror images of words and inverses of matrices. Mathematics Subject Classification. 68Q45, 68Q70.read more
Citations
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Abelian complexity of minimal subshifts
TL;DR: An affirmative answer to an old question of Rauzy is given by exhibiting a class of words whose Abelian complexity is everywhere equal to 3 by using van der Waerden's theorem, which shows that any minimal subshift having bounded Abelian simplicity contains Abelian k-powers for every positive integer k.
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Subword histories and Parikh matrices
TL;DR: This paper introduces the closely related notion of a subword history and obtains a sequence of general results: elimination of products, decidability of equivalence, and normal form, and investigates overall-methods for proving the validity of such results.
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Some characterizations of Parikh matrix equivalent binary words
S. Fossé,G. Richomme +1 more
TL;DR: It is shown that different characterizations of pair of words having the same Parikh matrix have different meanings according to the type of language they are spoken in.
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Connections between subwords and certain matrix mappings
TL;DR: This paper continues the investigation of Parikh matrices and subword occurrences, and studies certain inequalities, as well as information about subword occurrence sufficient to determine the whole word uniquely.
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Balance and Abelian complexity of the Tribonacci word
TL;DR: The Abelian complexity @r(n) of the Tribonacci word t which is the unique fixed point of @t is studied, it is shown that @r (n)@?{3,4,5,6,7} for each n>=1, and the word combinatorial properties of the generating morphism @t are exploited.
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Some Decision Problems Concerning Semilinearity and Commutation
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