Abelian Squares Research Articles

Counting abelian squares efficiently for a problem in quantum computing

Counting abelian squares efficiently for a problem in quantum computing

Arithmetic properties of Apéry-like numbers

We provide lower bounds for$p$-adic valuations of multisums of factorial ratios which satisfy an Apéry-like recurrence relation: these include Apéry, Domb and Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the$p$-adic valuation of Apéry numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the$p$-Lucas property for almost all primes$p$.

Abelian-square-rich words

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(n2) distinct factors, and there exist words of length n containing Θ(n2) 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 abelian-square-rich if, for every n, every factor of w of length n contains, on average, a number of distinct abelian-square factors that is quadratic in n; and uniformly abelian-square-rich if every factor of w contains a number of distinct abelian-square factors that is proportional to the square of its length. Of course, if a word is uniformly abelian-square-rich, then it is abelian-square-rich, but we show that the converse is not true in general. We prove that the Thue–Morse word is uniformly abelian-square-rich and that the function counting the number of distinct abelian-square factors of length 2n of the Thue–Morse word is 2-regular. As for Sturmian words, we prove that a Sturmian word sα of angle α is uniformly abelian-square-rich if and only if the irrational α has bounded partial quotients, that is, if and only if sα has bounded exponent.

Two triple binomial sum supercongruences

In a recent article, Apagodu and Zeilberger discuss some applications of an algorithm for finding and proving congruence identities (modulo primes) of indefinite sums of many combinatorial sequences. At the end, they propose some supercongruences as conjectures. Here we prove one of them, including a new companion enumerating Abelian squares, and we leave some remarks for the others.

Subquadratic-Time Algorithms for Abelian Stringology Problems

We propose the first subquadratic-time algorithms to a number of natural problems inabelian pattern matching (also called jumbled pattern matching) for strings over a constant-sized alphabet.Two strings are considered equivalent in this model if the numbers of occurrences of respectivesymbols in both of them, specified by their so-called Parikh vectors, are the same. We consider theproblems of computing abelian squares, abelian periods, abelian runs, abelian covers, and abelian borders.This work can be viewed as a continuation of a recent very active line of research on subquadraticspace and time jumbled indexing for binary and constant-sized alphabets (e.g., Moosa and Rahman,2012). Our algorithms apply the so-called four Russian technique in a fancy way and work in linearspace. The purpose of this work is to show that breaking the O(n2) barrier is possible for the aforementionedproblems. In this paper we extend our previous work, published in a preliminary form atMACIS 2015 conference, with effcient computation of abelian runs.

Maximum number of distinct and nonequivalent nonstandard squares in a word

The combinatorics of nonstandard squares in a word depends on how the equivalence of halves of the square is defined. We consider Abelian squares, parameterized squares, and order-preserving squares. The word uv is an Abelian (parameterized, order-preserving) square if u and v are equivalent in the Abelian (parameterized, order-preserving) sense. The maximum number of ordinary squares in a word is known to be asymptotically linear, but the exact bound is still investigated. We present several results on the maximum number of distinct squares for nonstandard factor equivalence relations. Let SQAbel(n,σ) and SQAbel′(n,σ) denote the maximum number of Abelian squares in a word of length n over an alphabet of size σ which are distinct as words and which are nonequivalent in the Abelian sense, respectively. For σ≥2 we prove that SQAbel(n,σ)=Θ(n2), SQAbel′(n,σ)=Ω(n3/2) and SQAbel′(n,σ)=O(n11/6). We also give linear bounds for parameterized and order-preserving squares for alphabets of constant size: SQparam(n,O(1))=Θ(n), SQop(n,O(1))=Θ(n). The upper bounds have quadratic dependence on the alphabet size for order-preserving squares and exponential dependence for parameterized squares. As a side result we construct infinite words over the smallest alphabet which avoid nontrivial order-preserving squares and nontrivial parameterized cubes (nontrivial parameterized squares cannot be avoided in an infinite word). A preliminary version of this paper was published at DLT 2014 [24]. In this full version we improve or extend the bounds on all three kinds of squares.

Existence of an infinite ternary 64-abelian square-free word

We consider a recently defined notion of k -abelian equivalence of words by concentrating on avoidance problems. The equivalence class of a word depends on the numbers of occurrences of different factors of length k for a fixed natural number k and the prefix of the word. We have shown earlier that over a ternary alphabet k -abelian squares cannot be avoided in pure morphic words for any natural number k . Nevertheless, computational experiments support the conjecture that even 3-abelian squares can be avoided over ternary alphabets. In this paper we establish the first avoidance result showing that by choosing k to be large enough we have an infinite k -abelian square-free word over three letter alphabet. In addition, this word can be obtained as a morphic image of a pure morphic word.

Abelian pattern avoidance in partial words

Pattern avoidance is an important topic in combinatorics on words which dates back to the beginning of the twentieth century when Thue constructed an infinite word over a ternary alphabet that avoids squares, i.e. , a word with no two adjacent identical factors. This result finds applications in various algebraic contexts where more general patterns than squares are considered. On the other hand, Erdős raised the question as to whether there exists an infinite word that avoids abelian squares, i.e. , a word with no two adjacent factors being permutations of one another. Although this question was answered affirmately years later, knowledge of abelian pattern avoidance is rather limited. Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. In this paper, we show that any pattern p with n > 3 distinct variables of length at least 2 n is abelian avoidable by a partial word with infinitely many holes, the bound on the length of p being tight. We complete the classification of all the binary and ternary patterns with respect to non-trivial abelian avoidability, in which no variable can be substituted by only one hole. We also investigate the abelian avoidability indices of the binary and ternary patterns.

On the average number of regularities in a word

We study the average number of powers and runs occurring in a word of length n drawn from an alphabet of size σ. We show that a word contains nσ(r−1)−1+o(n) powers of exponent r, at most nσ+o(n) runs, and also (1+2σ−1)n+o(n) palindromes. We also explore their abelian variants and prove that a binary word contains Θ(n32) abelian squares on the average, while its number of abelian cubes is related to Franel numbers. Finally, as a consequence, we show that a binary word has almost surely O(n32) abelian squares.

A note on efficient computation of all Abelian periods in a string

We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all “long” Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems.

Avoiding abelian squares in partial words

Erdős raised the question whether there exist infinite abelian square-free words over a given alphabet, that is, words in which no two adjacent subwords are permutations of each other. It can easily be checked that no such word exists over a three-letter alphabet. However, infinite abelian square-free words have been constructed over alphabets of sizes as small as four. In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. In particular, we give lower and upper bounds for the number of letters needed to construct infinite abelian square-free partial words with finitely or infinitely many holes. Several of our constructions are based on iterating morphisms. In the case of one hole, we prove that the minimal alphabet size is four, while in the case of more than one hole, we prove that it is five. We also investigate the number of partial words of length n with a fixed number of holes over a five-letter alphabet that avoid abelian squares and show that this number grows exponentially with n.

AVOIDING ABELIAN POWERS IN BINARY WORDS WITH BOUNDED ABELIAN COMPLEXITY

The notion of Abelian complexity of infinite words was recently used by the three last authors to investigate various Abelian properties of words. In particular, using van der Waerden's theorem, they proved that if a word avoids Abelian k-powers for some integer k, then its Abelian complexity is unbounded. This suggests the following question: How frequently do Abelian k-powers occur in a word having bounded Abelian complexity? In particular, does every uniformly recurrent word having bounded Abelian complexity begin in an Abelian k-power? While this is true for various classes of uniformly recurrent words, including for example the class of all Sturmian words, in this paper we show the existence of uniformly recurrent binary words, having bounded Abelian complexity, which admit an infinite number of suffixes which do not begin in an Abelian square. We also show that the shift orbit closure of any infinite binary overlap-free word contains a word which avoids Abelian cubes in the beginning. We also consider the effect of morphisms on Abelian complexity and show that the morphic image of a word having bounded Abelian complexity has bounded Abelian complexity. Finally, we give an open problem on avoidability of Abelian squares in infinite binary words and show that it is equivalent to a well-known open problem of Pirillo–Varricchio and Halbeisen–Hungerbühler.

Crucial abelian k-power-free words

Combinatorics In 1961, Erdos asked whether or not there exist words of arbitrary length over a fixed finite alphabet that avoid patterns of the form XX' where X' is a permutation of X (called abelian squares). This problem has since been solved in the affirmative in a series of papers from 1968 to 1992. Much less is known in the case of abelian k-th powers, i.e., words of the form X1X2 ... X-k where X-i is a permutation of X-1 for 2 <= i <= k. In this paper, we consider crucial words for abelian k-th powers, i. e., finite words that avoid abelian k-th powers, but which cannot be extended to the right by any letter of their own alphabets without creating an abelian k-th power. More specifically, we consider the problem of determining the minimal length of a crucial word avoiding abelian k-th powers. This problem has already been solved for abelian squares by Evdokimov and Kitaev (2004), who 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. Extending this result, we prove 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.

On shortest crucial words avoiding abelian powers

Let k ≥ 2 be an integer. An abelian k th power is a word of the form X 1 X 2 ⋯ X k where X i is a permutation of X 1 for 2 ≤ i ≤ k . A word W is said to be crucial with respect to abelian k th powers if W avoids abelian k th powers, but W x ends with an abelian k th power for any letter x occurring in W . Evdokimov and Kitaev (2004) [2] have shown that the shortest length of a crucial word on n letters avoiding abelian squares is 4 n − 7 for n ≥ 3 . Furthermore, Glen et al. (2009) [3] proved that this length for abelian cubes is 9 n − 13 for n ≥ 5 . They have also conjectured that for any k ≥ 4 and sufficiently large n , the shortest length of a crucial word on n letters avoiding abelian k th powers, denoted by ℓ k ( n ) , is k 2 n − ( k 2 + k + 1 ) . This is currently the best known upper bound for ℓ k ( n ) , and the best known lower bound, provided in Glen et al., is 3 k n − ( 4 k + 1 ) for n ≥ 5 and k ≥ 4 . In this note, we improve this lower bound by proving that for n ≥ 2 k − 1 , ℓ k ( n ) ≥ k 2 n − ( 2 k 3 − 3 k 2 + k + 1 ) ; thus showing that the aforementioned conjecture is true asymptotically (up to a constant term) for growing n .

Counting Abelian Squares

An abelian square is a nonempty string of length $2n$ where the last $n$ symbols form a permutation of the first $n$ symbols. Similarly, an abelian $r$'th power is a concatenation of $r$ blocks, each of length $n$, where each block is a permutation of the first $n$ symbols. In this note we point out that some familiar combinatorial identities can be interpreted in terms of abelian powers. We count the number of abelian squares and give an asymptotic estimate of this quantity.

A powerful abelian square-free substitution over 4 letters

In 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv , where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ 4 = { a , b , c , d } turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g 85 of Σ 4 ∗ . The size of this g 85 , i.e., | g 85 ( abcd ) | , is equal to 4 × 85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ 4 have been based on the structure of g 85 and on the endomorphism g 98 of Σ 4 ∗ found in 2002. In this paper, a great many new a-2-free endomorphisms of Σ 4 ∗ are reported. The sizes of these endomorphisms range from 4 × 102 to 4 × 115 . Importantly, twelve of the new a-2-free endomorphisms, of modulus m = 109 , can be used to construct an a-2-free (commutatively functional) substitution σ 109 of Σ 4 ∗ with 12 image words for each letter. The properties of σ 109 lead to a considerable improvement for the lower bound of the exponential growth of c n , i.e., of the number of a-2-free words over 4 letters of length n . It is obtained that c n > β − 50 β n with β = 1 2 1 / m ≃ 1.02306 . Originally, in 1998, Carpi established the exponential growth of c n by showing that c n > β − t β n with β = 2 19 / t = 2 19 / ( 8 5 3 − 85 ) ≃ 1.000021 , where t = 8 5 3 − 85 is the modulus of the substitution that he constructs starting from g 85 .

A One–Sided Zimin Construction

A string is Abelian square-free if it contains no Abelian squares; that is, adjacent substrings which are permutations of each other. An Abelian square-free string is maximal if it cannot be extended to the left or right by concatenating alphabet symbols without introducing an Abelian square. We construct Abelian square-free finite strings which are maximal by modifying a construction of Zimin. The new construction produces maximal strings whose length as a function of alphabet size is much shorter than that in the construction described by Zimin.

On Abelian squares and substitutions

We study substitutions preserving Abelian square-free words and the more general notion of substitutions with bounded Abelian squares. In particular, we prove the existence of algorithms deciding whether a substitution mapping each letter into a set of commutatively equivalent words belongs to one of these classes.

A characterization of the squares in a Fibonacci string

A (finite) Fibonacci string F n is defined as follows: F 0 = b, F 1 = a; for every integer n ⩾ 2, F n = F n − 1 F n − 2 . For n ⩾ 1, the length of F n is denoted by ƒ n = ¦F n¦ . The infinite Fibonacci string F is the string which contains every F n , n ⩾ 1, as a prefix. Apart from their general theoretical importance, Fibonacci strings are often cited as worst-case examples for algorithms which compute all the repetitions or all the “Abelian squares” in a given string. In this paper we provide a characterization of all the squares in F, hence in every prefix F n ; this characterization naturally gives rise to a Θ(ƒ n) algorithm which specifies all the squares of F n in an appropriate encoding. This encoding is made possible by the fact that the squares of F n occur consecutively, in “runs”, the number of which is Θ(ƒ n) . By contrast, the known general algorithms for the computation of the repetitions in an arbitrary string require Θ(ƒ n log ƒ n) time (and produce Θ(ƒ n log ƒ n) outputs) when applied to a Fibonacci string F n .