Generalized Thue-Morse words and palindromic richness extended abstract

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1 arxiv: v2 [math.co] 26 Apr Introduction Generalized Thue-Morse words and palindromic richness extended abstract Štěpán Starosta Department of Mathematics, FNSPE, Czech Technical University in Prague, Trojanova 13, Praha 2, Czech Republic Palindromic richness is a very intensively studied notion. For infinite words with language closed under reversal there exist many equivalent characterizations of palindromic richness. Each of them can be adopted as a definition of palindromic richness. Let us list three of these characterizations. An infinite word u is rich if one of the following holds: 1. any factor w of u of length n contains exactly n+1 palindromic factors; 2. each complete return word of any palindrome occurring in u is a palindrome as well; 3. for any n N, the equality C(n) + 2 = P(n) + P(n + 1) is satisfied, where C(n) denotes the first increment of factor complexity of u and P(n) denotes the palindromic complexity of u. Let us mention that factor of any infinite word u of length n contains at most n+1 palindromic factors, and the inequality C(n)+2 P(n)+P(n+1) (1) is valid for any infinite word u with language closed under reversal and for any n. Therefore, the fact that a word u is rich in palindromes expresses that u is saturated by palindromes up to the highest possible level. Arnoux-Rauzy words (among them Sturmian words) and words coding interval exchange transformation with the symmetric permutation belong to the most prominent examples of rich words. Palindrome is a word which coincides with its mirror image, or more formally, w = Θ 0 (w), where w is a finite word and Θ 0 : A A is defined by Θ 0 (w 1 w 2...w n ) = w n w n 1...w 1 for lettersw i A. If we replace the mirror mapping Θ 0 by an antimorphism Θ, we can define Θ-palindromes as words which are fixed points of Θ, i.e., w = Θ(w). For any antimorphism Θ, the notion of Θ-palindromic richness can be introduced and 1

2 an analogy of characterizations 1, 2 and 3 mentioned above can be formulated, see [11]. Although the language of the Thue-Morse word is closed under two antimorphisms, it is not Θ-rich for any of these two antimorphisms. The author and E. Pelantová focused in [9] on infinite words u with language invariant under more antimorphisms simultaneously. For a given finite group G formed by morphisms and antimorphisms ona, the words with language invariant under any element of G are investigated. If, moreover, such a word u is uniformly recurrent, then a generalized version of the inequality (1) is proven: there exists an integer N such that C(n)+#G ( ) P Θ (n)+p Θ (n+1) for any n N, (2) Θ G (2) where G (2) denotes the set of involutive antimorphisms of G and P Θ is the Θ-palindromic complexity, i.e., P Θ (n) counts the number of Θ-palindromes of length n in the language of u. Infinite uniformly recurrent words with language closed under all elements of G and for which the equality in (2) is attained for any n N are called almost G-rich. Let us emphasize that in case when the group G contains besides identity only the mirror mapping Θ 0, the notion of almost richness (as introduced in [8]) and the notion of almost G-richness coincide. Let us mention that in [9] also G-richness is introduced. In case of G = {Θ 0,id} it coincides with classical palindromic richness. The definition requires further notions, thus we will omit it here and deal only with almost G-richness. In this article we concentrate on the generalized Thue-Morse sequences t b,m already considered by E. Prouhet in 1851, see [10]. Let s b (n) denote the sum of digits in the base-b representation of the integer n, for integers b 2 and m 1. The infinite word t b,m is defined as t b,m = (s b (n) mod m) + n=0. Using this notation, the famous Thue-Morse word equals t 2,2. The word t b,m is over the alphabet {0,1,...,m 1} = Z m. Similarly to the classical Thue-Morse word, also t b,m is a fixed point of a primitive substitution, as already mentioned in [1]. It is easy to see that the substitution defined by ϕ b,m (k) = k(k +1)(k +2)... ( k +(b 1) ) for any k Z m fixes the word t b,m. As already stated in [1], it can be shown that t b,m is periodic if and only if b 1 mod m. We show that for any parameters b and m the language of the word t b,m is invariant under all elements of the dihedral group of order 2m, denoted D m, see Proposition 1, and that t b,m is almost D m -rich, see Theorem 5. 2 Preliminaries We will use common notions from combinatorics on words. By A we denote an alphabet. For a one-sided infinite word u, we denote by L(u) its set of factors (called the language 2

3 of u), by L n (u) we denote its factors of length n. By C(n) we denote the factor complexity of u. A letter a A is a left extension of a factor w L(u) if aw L(u). If a factor has at least two distinct left extensions, then it is said to be left special. The set of all left extensions of w is denoted Lext(w). The definition of right extensions, Rext(w) and right special is analogous. A factor which is left and right special is called bispecial (BS). The bilateral order b(w) of a factor w is the number b(w) := #Bext(w) #Lext(w) #Rext(w)+1 where the quantity Bext(w) = {awb L(u) a,b A}. In [5], the following relation between the second increment of factor complexity and bilateral orders is shown: C(n+2) 2C(n+1)+C(n) = 2 C(n) = b(w). (3) w L n(u) One can easily show that if w is not a bispecial factor, then b(w) = 0. Thus, to enumerate the factor complexity of an infinite word u, one needs to calculate C(0), C(1), and bilateral orders of all its bispecial factors. A mapping ϕ on A is called a morphism if ϕ(vw) = ϕ(v)ϕ(w) for any v,w A ; an antimorphism if ϕ(vw) = ϕ(w)ϕ(v) for any v,w A. By AM(A ) we denote the set of all morphisms and antimorphisms over A. Let ν AM(A ). We say that a word u is closed under ν if for all w L(u) we have ν(w) L(u). It is clear that the reversal mapping Θ 0 is an antimorphism. Moreover, it is an involution, i.e., Θ 2 0 = id. A fixed point of a given antimorphism Θ is called Θ-palindrome. If Θ = Θ 0, then we say palindrome or classical palindrome instead of Θ 0 -palindrome. The set of all Θ-palindromic factors of an infinite word u is denoted by Pal Θ (u). The Θ-palindromic complexity of u is the mapping P Θ : N N given by P Θ (n) = #(Pal Θ (u) L n (u)). If a A, w Pal Θ (u), and awθ(a) L(u), then awθ(a) is said to be a Θ-palindromic extension of w in u. The set of all Θ-palindromic extensions of w is denoted by Pext Θ (w). 3 Generalized Thue-Morse words and dihedral groups In this section we will show that t b,m is almost G-rich for a concrete group G AM(A ). Fix b and m. To abbreviate, we denote ϕ = ϕ b,m. In what follows, the alphabet will be considered Z m, and letters will be expressed modulo m to ease the notation. For all x Z m denote by Ψ x the antimorphism given by and by Π x the morphism given by Ψ x (k) := x k for all k Z m Π x (k) := x+k for all k Z m. Denote by D m the set D m = {Ψ x x Zm } {Π x x Zm }. It is easy to show that D m can be generated by only 2 elements, for instance one can choose Π 1 and Ψ 0. In fact, D m is exactly the dihedral group of order 2m. D m is exactly the group such that t b,m is closed under all its elements. 3

4 Proposition 1. The word t b,m is closed under all elements of D m. Proof. It is easy to show that for all x Z m, Π x ϕ = ϕπ x. Furthermore, one can also show Ψ x ϕ = ϕψ x+b 1. Using these two relations and inducing on n, one can show that for all ν D m, w L n (t b,m ) ν(w) L n (t b,m ). 4 Almost D m -richness of t b,m In this section we will show that the word t b,m is almost D m -rich. Since one has to proceed differently in the periodic case, we consider only the aperiodic case, i.e., b 1 mod m. To simplify the following, we restrict ourselves to b 1 and m coprime. We will say that a factor v = v 0 v 1...v s 1 L(t b,m ) is an ancestor of a factor w L(t b,m ) if w is a factor of ϕ(v) and is not a factor of ϕ(v 1...v s 1 ) or ϕ(v 0...v s 2 ). The following properties of ϕ and t b,m can be easily deduced. 1. ϕ is uniform, i.e., for all k,l Z m, ϕ(k) = ϕ(l). 2. ϕ is bifix-free, i.e., for all k,l Z m, the first letter of ϕ(k) differs from the first letter of ϕ(l), and so does the last letter. 3. L 2 (t b,m ) = {tr t,r Z m }. 4. L 3 (t b,m ) = {(t 1)tr t,r Z m } {rt(t+1) t,r Z m }. 5. If for x Z m, the word w L(t b,m ) is a Ψ x -palindrome, then the factor ϕ(w) is a Ψ x b+1 -palindrome. 6. Let w L(t b,m ). w is BS factor ν(w) is BS factor for all ν D m. Furthermore, b(w) = b(ν(w)) and if w is a Θ-palindrome for some antimorphism Θ D m, then ν(w) is a Θ -palindrome for some Θ D m. 7. If w = w 0...w s 1 L(t b,m ) and there is an index i such that w i+1 w i +1, then w has exactly one ancestor. 8. If w L(t b,m ) and w 2b+1, then w has exactly one ancestor. 9. If w L(t b,m ) is BS and w b, then there exist letters x and y such that ϕ(x) is a prefix of w and ϕ(y) is a suffix of w. These properties are used to prove the next two lemmas. The first summarizes the bilateral orders and Θ-palindromic extensions of longer BS factors. Lemma 2. Let b 1 and m be coprime and b 1 mod m. Let w L(t b,m ) be a BS factor such that w 2b. Then there exists a BS factor v such that ϕ(v) = w. Furthermore, b(w) = b(v). If v is a Θ 1 -palindrome, Θ 1 D m, then there exists a unique Θ 2 D m such that w is a Θ 2 -palindrome. Moreover, #Pext Θ2 (w) = #Pext Θ1 (v). 4

5 Thanks to the last lemma, we have to evaluate only the bilateral orders and number of palindromic extensions of shorter factors. The next lemma exhibits these values for concerned lengths of BS factors. Lemma 3. Let b 1 and m be coprime and b 1 mod m. Let w be a BS factor of t b,m such that 1 w 2b 1. Let Θ D m be the unique antimorphism such that w = Θ(w). Then the values b(w) and #Pext Θ (w) are as shown in Table 1. w b(w) #Pext Θ (w) 1 w b w = b 1 2 b+1 w 2b w = 2b Table 1: The bilateral order and number of palindromic extensions of a BS factor w of t b,m, b 1 mod m, according to its length w. Θ D m is the antimorphism such that Θ(w) = w. Corollary 4. Let w be a non-empty BS factor of t b,m, b 1 and m be coprime, and b 1 mod m. Then 1. there exists a unique antimorphism Θ D m such that Θ(w) = w; 2. b(w) = #Pext Θ (w) 1. Theorem 5. The word t b,m is almost D m -rich. Proof. First, let b 1 mod m. We will show that ( ) C(n)+2m = P Θ (n)+p Θ (n+1) for all n 1. (4) Θ D m Θ antimorphism Note that #D m = 2m. First, we will show the relation (4) for n = 1. It is clear that C(1) = m and C(2) = m 2. Thus, the left-hand side equals m 2 m+2m = m 2 +m. If m = 2q 1, then for all x Z m, P Ψx (1) = 1. If m = 2q, then for all y Z m, P Ψ2y (1) = 2 and P Ψ2y+1 (1) = 0. Furthermore, one can show that for all antimorphism Θ D m, we have P Θ (2) = m. Therefore, the right-hand side equals m 2 +m. To show the relation (4) for n+1, where n > 1, we will verify that the difference of the left-hand sides for indices n+1 and n equals the difference of the right-hand sides for the same indices. In other words, we are going to show that ( ) C(n+1) C(n) = P Θ (n+2) P Θ (n). (5) Θ D m Θ antimorphism 5

6 According to the equation (3), the left-hand side side can be written as 2 C(n) = b(w) and for the right-hand side we can use w L n(t b,m ) w BS P Θ (n+2) P Θ (n) = w L n(t b,m ) w=θ(w) ( #PextΘ (w) 1 ). Finally, using the fact that a non-bispecial Θ-palindrome has exactly one Θ-palindromic extension and Corollary 4, the relation (5) holds. If b 1 and m are coprime, minor modifications have to be made to above statements and also to the proofs. If t b,m is periodic, i.e., b 1 mod m, the proof of almost D m -richness can be done in a very similar way and is left to the reader. Remark 6. In fact, the generalized Thue-Morse words are D m -rich, not only almost D m -rich. To show the D m -richness, one has to use the definition of G-richness which is based on a specific graph Γ n (t b,m ) representing L n (t b,m ) (for details see [9]). These graphs can be used to study the densities of factors of length n + 1 using the same method as in [2]. The densities of factors of a certain class of words including the Thue-Morse word were explored in [7]. Remark 7. It is clear that using (3) and Lemmas 2 and 3, one could evaluate the factor complexity and Θ-palindromic complexities of t b,m for all b and m. To our knowledge, the factor complexity of the Thue-Morse sequence t 2,2 was described in 1989 independently in [4] and [6] and of t 2,m in [12]. The palindromic complexities of the Thue-Morse word were described in [3]. Acknowledgments This work was supported by the Czech Science Foundation grant GAČR 201/09/0584, by the grants MSM and LC06002 of the Ministry of Education, Youth, and Sports of the Czech Republic, and by the grant of the Grant Agency of the Czech Technical University in Prague grant No. SGS11/162/OHK4/3T/14. References [1] J.-P. Allouche and J. Shallit, Sums of digits, overlaps, and palindromes, Discrete Math. Theoret. Comput. Sci., 4 (2000), pp [2] L. Balková and E. Pelantová, A note on symmetries in the rauzy graph and factor frequencies, Theoret. Comput. Sci., 410 (2009), pp

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