160 research outputs found
Abelian complexity of fixed point of morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1
We study the combinatorics of vtm, a variant of the Thue-Morse word generated by the non-uniform morphism 0 ↦ 012, 1 ↦ 02, 2 ↦ 1 starting with 0. This infinite ternary sequence appears a lot in the literature and finds applications in several fields such as combinatorics on words; for example, in pattern avoidance it is often used to construct infinite words avoiding given patterns. It has been shown that the factor complexity of vtm, i.e., the number of factors of length n, is Θ(n); in fact, it is bounded by ¹⁰⁄₃n for all n, and it reaches that bound precisely when n can be written as 3 times a power of 2. In this paper, we show that the abelian complexity of vtm, i.e., the number of Parikh vectors of length n, is O(log n) with constant approaching ¾ (assuming base 2 logarithm), and it is Ω(1) with constant 3 (and these are the best possible bounds). We also prove some results regarding factor indices in vtm."F. Blanchet-Sadri and Nathan Fox’s research was supported by the National Science Foundation under Grant No. DMS–1060775."
"James D. Currie and Narad Rampersad’s research was supported by NSERC Discovery grants.
Avoiding 2-binomial squares and cubes
Two finite words are 2-binomially equivalent if, for all words of
length at most 2, the number of occurrences of as a (scattered) subword of
is equal to the number of occurrences of in . This notion is a
refinement of the usual abelian equivalence. A 2-binomial square is a word
where and are 2-binomially equivalent.
In this paper, considering pure morphic words, we prove that 2-binomial
squares (resp. cubes) are avoidable over a 3-letter (resp. 2-letter) alphabet.
The sizes of the alphabets are optimal
A new approach to the -regularity of the -abelian complexity of -automatic sequences
We prove that a sequence satisfying a certain symmetry property is
-regular in the sense of Allouche and Shallit, i.e., the -module
generated by its -kernel is finitely generated. We apply this theorem to
develop a general approach for studying the -abelian complexity of
-automatic sequences. In particular, we prove that the period-doubling word
and the Thue--Morse word have -abelian complexity sequences that are
-regular. Along the way, we also prove that the -block codings of these
two words have -abelian complexity sequences that are -regular.Comment: 44 pages, 2 figures; publication versio
Relations on words
In the first part of this survey, we present classical notions arising in combinatorics on words: growth function of a language, complexity function of an infinite word, pattern avoidance, periodicity and uniform recurrence. Our presentation tries to set up a unified framework with respect to a given binary relation.
In the second part, we mainly focus on abelian equivalence, -abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and -equivalence. In particular, some new refinements of abelian equivalence are introduced
The Simplest Binary Word with Only Three Squares
We re-examine previous constructions of infinite binary words containing few
distinct squares with the goal of finding the "simplest", in a certain sense.
We exhibit several new constructions. Rather than using tedious case-based
arguments to prove that the constructions have the desired property, we rely
instead on theorem-proving software for their correctness
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