A word is "crucial" with respect to a given set of "prohibited words" (or simply "prohibitions") if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A "minimal crucial word" is a crucial word of the shortest length. A word W contains an "abelian k-th power" if W has a factor of the form X_1X_2...X_k where X_i is a permutation of X_1 for 2<= i <= k. When k=2 or 3, one deals with "abelian squares" and "abelian cubes", respectively.
In 2004 (arXiv:math/0205217), Evdokimov and Kitaev showed that a minimal crucial word over an n-letter alphabet A_n = {1,2,..., n} avoiding abelian squares has length 4n-7 for n >= 3. In this paper we show that a minimal crucial word over A_n avoiding abelian cubes has length 9n-13 for n >= 5, and it has length 2, 5, 11, and 20 for n=1, 2, 3, and 4, respectively. Moreover, for n >= 4 and k >= 2, we give a construction of length k^2(n-1)-k-1 of a crucial word over A_n avoiding abelian k-th powers. This construction gives the minimal length for k=2 and k=3. For k >= 4 and n >= 5, we provide a lower bound for the length of crucial words over A_n avoiding abelian k-th powers.
We show that the 2-abelian complexity of the infinite Thue-Morse word is 2-regular, and other properties of the 2-abelian complexity, most notably that it is a concatenation of palindromes of increasing length. We also show sharp bounds for the length of unique extensions of factors of size n, occurring in the Thue-Morse word.
We revisit the question of classification of balanced circular words and focus on the case of a ternary alphabet. We propose a $3$-dimensional generalisation of the discrete approximation representation of Christoffel words. By considering the minimal bound $3$ for abelian complexity of balanced circular words over a ternary alphabet, we provide a classification of all circular words over a ternary alphabet with abelian complexity subject to this bound. This result also allows us to construct an uncountable set of bi-infinite aperiodic words with abelian complexity equal to $3$.
We study the following problem, first introduced by Dekking. Consider an infinite word x over an alphabet {0,1,...,k-1} and a semigroup homomorphism S:{0,1,...,k-1}* -> N. Let L_x denote the set of factors of x. What conditions on S and the abelian complexity of x guarantee that S(L_x) contains all but finitely many elements of N? We examine this question for some specific infinite words x having different abelian complexity functions.
In this paper, we study the relation between periodicity of two-dimensional words and their abelian pattern complexity. A pattern $\cal{P}$ in $\mathbb{Z}^n$ is the set of all translations of some finite subset $F$ of $\mathbb{Z}^n$. An $F$-factor of an infinite word is a finite word restricted to $F$. Then the pattern complexity over a pattern $\mathcal{P}$ counts the number of distinct $F$-factors of an infinite word, for $P\in \mathcal{P}$. Two finite words are called abelian equivalent if for each letter of the alphabet, they contain the same numbers of occurrences of this letter. The abel…
Two finite words $u$ and $v$ are called abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of an infinite word $\mathbf{x}$ is the set of infinite words $\mathbf{y}$ such that, for each factor $u$ of $\mathbf{y}$, there exists a factor $v$ of $\mathbf{x}$ which is abelian equivalent to $u$. The notion of an abelian closure gives a characterization of Sturmian words: among uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which $\mathcal{A}(\mathbf{x})$ equals the shift …
Two finite words $u$ and $v$ are called Abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of (the shift orbit closure of) an infinite word $\mathbf{x}$ is the set of infinite words $\mathbf{y}$ such that, for each factor $u$ of $\mathbf{y}$, there exists a factor $v$ of $\mathbf{x}$ which is abelian equivalent to $u$. The notion of an abelian closure gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which $\mathcal{A}(\mathbf{x})$ equals the …
We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine if certain types of morphic sequences avoid additive powers. We also show that there are $Ω(1.172^n)$ binary words of length $n$ that avoid abelian 4-powers, which improves on previous estimates.
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain $Θ(n^2)$ distinct factors that are abelian squares. We study infinite words such that the number of abelian square factors of length $n$ grows quadratically with $n$.
An abelian anti-power of order $k$ (or simply an abelian $k$-anti-power) is a concatenation of $k$ consecutive words of the same length having pairwise distinct Parikh vectors. This definition generalizes to the abelian setting the notion of a $k$-anti-power, as introduced in [G. Fici et al., Anti-powers in infinite words, J. Comb. Theory, Ser. A, 2018], that is a concatenation of $k$ pairwise distinct words of the same length. We aim to study whether a word contains abelian $k$-anti-powers for arbitrarily large $k$. S. Holub proved that all paperfolding words contain abelian powers of every o…
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these res…
Abelian maximal pattern complexity of two-dimensional words
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Information Science, Shandong Technology and Business University（山东工商学院数学与信息科学学院）
An abelian square is the concatenation of two words that are anagrams of one another. A word of length $n$ can contain at most $Θ(n^2)$ distinct factors, and there exist words of length $n$ containing $Θ(n^2)$ distinct abelian-square factors, that is, distinct factors that are abelian squares. This motivates us to study infinite words such that the number of distinct abelian-square factors of length $n$ grows quadratically with $n$. More precisely, we say that an infinite word $w$ is {\it abelian-square-rich} if, for every $n$, every factor of $w$ of length $n$ contains, on average, a number o…
We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are Mäkelä's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite binary words, and therefore answering negatively one question from Mäkelä. Then we show that one can avoid $3$-abelian-squares of period at least $3$ in infinite binary words and $2$-abelian-squares of period at least 2 in infinite ternary words. Finally we study the minimum number of distinct $k$-abelian-squares that must appear in an infinite binary word.
The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $θ$ there exists an infinite word having abelian critical exponent $θ$. We also extend this result to the $k$…
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, $k$-abelian equivalence, combinatorial coefficients and associated relations, Parikh matrices and $M$-equivalence. In particular, some new refinements of abelian equivalence are introduced.
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $α$. vAs an analogue of the critical exponent, we introduce the abelian critical …
According to a result of Richomme, Saari and Zamboni, the abelian complexity of the Tribonacci word satisfies $ρ^{\mathrm{ab}}(n)\in\{3,4,5,6,7\}$ for each $n\in\mathbb{N}$. In this paper we derive an automaton that evaluates the function $ρ^{\mathrm{ab}}(n)$ explicitly. The automaton takes the Tribonacci representation of $n$ as its input; therefore, $(ρ^{\mathrm{ab}}(n))_{n\in\mathbb{N}}$ is an automatic sequence in a generalized sense. Since our evaluation of $ρ^{\mathrm{ab}}(n)$ uses $\mathcal{O}(\log n)$ operations, it is fast even for large values of $n$. Our result also leads to a solut…
Abelian maximal pattern complexity and extremal words
School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematics and Information Science, Shandong Technology and Business University（山东工商学院数学与信息科学学院）
Two words $u$ and $v$ are $k$-abelian equivalent if, for each word $x$ of length at most $k$, $x$ occurs equally many times as a factor in both $u$ and $v$. The notion of $k$-abelian equivalence is an intermediate notion between the abelian equivalence and the equality of words. In this paper, we study the equivalence classes induced by the $k$-abelian equivalence, mainly focusing on the cardinalities of the classes. In particular, we are interested in the number of singleton $k$-abelian classes, i.e., classes containing only one element. We find a connection between the singleton classes and …
We study the set of $k$-abelian critical exponents of all Sturmian words. It has been proven that in the case $k = 1$ this set coincides with the Lagrange spectrum. Thus the sets obtained when $k > 1$ can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when $k > 1$ the spectrum is a dense non-closed set. This is in contrast with the case $k = 1$, where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra.…
We present a general method for computing the abelian complexity $ρ^{\rm ab}_{\bf s} (n)$ of an automatic sequence $\bf s$ in the case where (a) $ρ^{\rm ab}_{\bf s} (n)$ is bounded by a constant and (b) the Parikh vectors of the length-$n$ prefixes of $\bf s$ form a synchronized sequence. We illustrate the idea in detail, using the free software Walnut to compute the abelian complexity of the Tribonacci word ${\bf TR} = 0102010\cdots$, the fixed point of the morphism $0 \rightarrow 01$, $1 \rightarrow 02$, $2 \rightarrow 0$. Previously, Richomme, Saari, and Zamboni showed that the abelian comp…
