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Abelian Square-Free Partial Words
[chapter]
2010
Lecture Notes in Computer Science
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Infinite abelian square-free words have been constructed over alphabets of sizes as small as four. ...
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. ...
The partial word w is abelian square-free if it does not have any full or partial abelian square, except those of the form a or a , where a ∈ A. ...
doi:10.1007/978-3-642-13089-2_8
fatcat:rq3duhwdwfefhdns37oivwckbq
On the number of partially abelian square-free words on a three-letter alphabet
1991
Theoretical Computer Science
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Let # be a &square-free morphism, consider any square-free word f of length not greater than 5; f is also &square-free, hence e(f) is square-free. ...
El A morphism # from {a, b, c}* into itself is &square-free if, for any &square-free word f, e(f) is also &square-free. ...
doi:10.1016/0304-3975(91)90322-s
fatcat:tehmhj3z4fe65cmmsz3ecdvhuy
Avoiding abelian squares in partial words
2012
Journal of combinatorial theory. Series A
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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. ...
In this paper, we investigate the problem of avoiding abelian squares in partial words, or sequences that may contain some holes. ...
Robert Merca¸s' work was partially supported by Research Grant No. 1323 U07 E30 N-2008/InvAct/Bel, G. ...
doi:10.1016/j.jcta.2011.08.008
fatcat:5sarssevh5ao3iv2mcyriurudq
Abelian repetitions in partial words
2012
Advances in Applied Mathematics
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Then we show, in a number of cases, that the number of abelian p-free partial words of length n with h holes over a given alphabet grows exponentially as n increases. ...
We study abelian repetitions in partial words, or sequences that may contain some unknown positions or holes. ...
The partial word u is abelian square-free if it does not have any full or partial abelian square, except those of the form a or a , where a ∈ A. ...
doi:10.1016/j.aam.2011.06.006
fatcat:cx2docar75hehmfhjxoaqnclym
Abelian Pattern Avoidance in Partial Words
[chapter]
2012
Lecture Notes in Computer Science
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Recently, (abelian) pattern avoidance was initiated in the more general framework of partial words, which allow for undefined positions called holes. ...
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. ...
A partial word is abelian square-free if it does not contain any factor that results in an abelian square after filling in the holes. ...
doi:10.1007/978-3-642-32589-2_21
fatcat:25ohmrc7lrgchgragwi5pje67y
Words with the Maximum Number of Abelian Squares
[article]
2015
arXiv
pre-print
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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. ...
We also proved that Sturmian words that are β-power free for some β ≥ 2 are uniformly abelian-square rich. ...
arXiv:1506.03562v1
fatcat:i2fgs6ze6zh6jfpuau3e5dpxqq
A powerful abelian square-free substitution over 4 letters
2009
Theoretical Computer Science
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An abelian square is a non-empty word uv, where u and v are permutations (anagrams) of each other. ...
In 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. ...
Acknowledgements Originally, the author in [42] presented the main results of this paper at WORDS'07, the 6th International Conference on Combinatorics on Words, Institut de Mathématiques de Luminy, ...
doi:10.1016/j.tcs.2009.05.027
fatcat:p2orhjifxnfnzmy7wwc3i5lnw4
5-Abelian cubes are avoidable on binary alphabets
2014
RAIRO - Theoretical Informatics and Applications
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For example, the word banana contains the squares anan and nana and the overlap anana. A ...
For a word uvw, we say that uv is a square if u = v and both words are non-empty, and we say that uvw is an overlap if all words are non-empty, and either u = v and w is a prefix of u, or v = w and u is ...
For example, intestines is an abelian square where intes is a permutation of the word tines. A word is abelian square-free, if it does not contain any abelian square as a factor. ...
doi:10.1051/ita/2014020
fatcat:mvooknvgpzhwxglv7w3gbukcc4
Pattern avoidance: themes and variations
2005
Theoretical Computer Science
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We review results concerning words avoiding powers, abelian powers or patterns. In addition we collect/pose a large number of open problems. ...
The study of abelian squares was initiated by Erdös [38] . As in the non-abelian case, the study of abelian square-free words has algebraic interpretations (see [46] ). ...
Thus a square-free word avoids xx. More generally, a word which avoids An abelian square is a word x 1 x 2 where x 2 can be obtained from x 1 by rearranging letters. ...
doi:10.1016/j.tcs.2005.01.004
fatcat:cr5bmro63zbbnco3ogmsuw7yly
Abelian-square-rich words
2017
Theoretical Computer Science
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2017 pre-print arXiv:1701.00948v2 fatcat:jh5apbe7dnat3imhcydpzlv674Other Versions
As for Sturmian words, we prove that a Sturmian word s_α 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 s_α has bounded ...
An abelian square is the concatenation of two words that are anagrams of one another. ...
Now we prove that if α has bounded partial quotients, then the Sturmian word s α is (uniformly) abelian-square-rich. ...
doi:10.1016/j.tcs.2017.02.012
fatcat:kvuemu6aw5djvkvyzv7hqghcce
On Abelian squares and substitutions
1999
Theoretical Computer Science
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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. ...
The Abelian square-free words exhibited in the quoted papers are obtained by iteration of morphisms preserving Abelian square-free words (Abelian square-free morphisms). ...
doi:10.1016/s0304-3975(98)00250-3
fatcat:tfxadtagmreblie2rimyuceij4
Avoidability of long k-abelian repetitions
[article]
2015
arXiv
pre-print
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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. ...
Then h 4 is (3, 3)-abelian-square-free. Moreover, for every abelian-square-free word w, h 4 (w) contains only 4 distinct 3-abelian-squares: 0 2 , 1 2 , (01) 2 and (10) 2 . ...
arXiv:1507.02581v1
fatcat:bvyju3mcivfmjl2iwk57gyrgdm
Square-Free Words over Partially Commutative Alphabets
[chapter]
2015
Lecture Notes in Computer Science
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We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word v in the case of a partially commutative alphabet of size three. ...
We give two characterizations (morphic and recursive) of the lexicographically minimal square-free word v in the case of a partially commutative alphabet Θ of size three. ...
Square-free and abelian square-free words have been extensively studied. ...
doi:10.1007/978-3-319-15579-1_33
fatcat:fqfho7swnrbilj4j3cpyhsczba
Partially abelian squarefree words
1990
RAIRO - Theoretical Informatics and Applications
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PARTIALLY ABELIAN SQUARE FREE WORDS IN A FOUR LETTERS ALPHABET In this section, A is the alphabet {a, b, c, d}. ...
PARTIALLY ABELIAN SQUARE FREE WORDS IN { a, b, c }* In this section A is the alphabet consisting of the three letters {a, b, c} and 0 t is the relation consisting of the two pairs {{a, c), (c, a)}, 0 2 ...
doi:10.1051/ita/1990240605091
fatcat:tyvyqptzvbdtxk4c56du3tqqmy
3-Abelian Cubes Are Avoidable on Binary Alphabets
[chapter]
2013
Lecture Notes in Computer Science
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A k-abelian cube is a word uvw, where u, v, w have the same factors of length at most k with the same multiplicities. ...
Previously it has been known that k-abelian cubes are avoidable over a binary alphabet for k ≥ 5. Here it is proved that this holds for k ≥ 3. ...
Computer experiments would suggest that there are 3-abelian square-free ternary words, but it is known that there are no pure morphic k-abelian square-free ternary words for any k [13] . ...
doi:10.1007/978-3-642-38771-5_33
fatcat:7povcv5jujghrjjhplhpvx3vs4
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