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Abelian Square-Free Partial Words
[chapter]

2010
*
Lecture Notes in Computer Science
*

Infinite

doi:10.1007/978-3-642-13089-2_8
fatcat:rq3duhwdwfefhdns37oivwckbq
*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. ...##
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On the number of partially abelian square-free words on a three-letter alphabet

1991
*
Theoretical Computer Science
*

Let # be a &

doi:10.1016/0304-3975(91)90322-s
fatcat:tehmhj3z4fe65cmmsz3ecdvhuy
*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*. ...##
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Avoiding abelian squares in partial words

2012
*
Journal of combinatorial theory. Series A
*

In particular, we give lower and upper bounds for the number of letters needed to construct infinite

doi:10.1016/j.jcta.2011.08.008
fatcat:5sarssevh5ao3iv2mcyriurudq
*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. ...##
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Abelian repetitions in partial words

2012
*
Advances in Applied Mathematics
*

Then we show, in a number of cases, that the number of

doi:10.1016/j.aam.2011.06.006
fatcat:cx2docar75hehmfhjxoaqnclym
*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. ...##
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Abelian Pattern Avoidance in Partial Words
[chapter]

2012
*
Lecture Notes in Computer Science
*

Recently, (

doi:10.1007/978-3-642-32589-2_21
fatcat:25ohmrc7lrgchgragwi5pje67y
*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. ...##
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Words with the Maximum Number of Abelian Squares
[article]

2015
*
arXiv
*
pre-print

An

arXiv:1506.03562v1
fatcat:i2fgs6ze6zh6jfpuau3e5dpxqq
*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. ...##
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A powerful abelian square-free substitution over 4 letters

2009
*
Theoretical Computer Science
*

An

doi:10.1016/j.tcs.2009.05.027
fatcat:p2orhjifxnfnzmy7wwc3i5lnw4
*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, ...##
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5-Abelian cubes are avoidable on binary alphabets

2014
*
RAIRO - Theoretical Informatics and Applications
*

For example, the

doi:10.1051/ita/2014020
fatcat:mvooknvgpzhwxglv7w3gbukcc4
*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. ...##
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Pattern avoidance: themes and variations

2005
*
Theoretical Computer Science
*

We review results concerning

doi:10.1016/j.tcs.2005.01.004
fatcat:cr5bmro63zbbnco3ogmsuw7yly
*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. ...##
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Abelian-square-rich words

2017
*
Theoretical Computer Science
*

As for Sturmian

doi:10.1016/j.tcs.2017.02.012
fatcat:kvuemu6aw5djvkvyzv7hqghcce
*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. ...##
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On Abelian squares and substitutions

1999
*
Theoretical Computer Science
*

We study substitutions preserving

doi:10.1016/s0304-3975(98)00250-3
fatcat:tfxadtagmreblie2rimyuceij4
*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). ...##
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Avoidability of long k-abelian repetitions
[article]

2015
*
arXiv
*
pre-print

Then we show that one can avoid 3-

arXiv:1507.02581v1
fatcat:bvyju3mcivfmjl2iwk57gyrgdm
*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 . ...##
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Square-Free Words over Partially Commutative Alphabets
[chapter]

2015
*
Lecture Notes in Computer Science
*

We give two characterizations (morphic and recursive) of the lexicographically minimal

doi:10.1007/978-3-319-15579-1_33
fatcat:fqfho7swnrbilj4j3cpyhsczba
*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. ...##
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Partially abelian squarefree words

1990
*
RAIRO - Theoretical Informatics and Applications
*

*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 ...

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3-Abelian Cubes Are Avoidable on Binary Alphabets
[chapter]

2013
*
Lecture Notes in Computer Science
*

A k-

doi:10.1007/978-3-642-38771-5_33
fatcat:7povcv5jujghrjjhplhpvx3vs4
*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] . ...
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