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Smooth words on 2-letter alphabets having same parity

Identifieur interne : 000318 ( PascalFrancis/Corpus ); précédent : 000317; suivant : 000319

Smooth words on 2-letter alphabets having same parity

Auteurs : S. Brlek ; D. Jamet ; G. Paquin

Source :

RBID : Pascal:08-0195807

Descripteurs français

English descriptors

Abstract

In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b are odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b-1+1).

Notice en format standard (ISO 2709)

Pour connaître la documentation sur le format Inist Standard.

pA  
A01 01  1    @0 0304-3975
A02 01      @0 TCSCDI
A03   1    @0 Theor. comput. sci.
A05       @2 393
A06       @2 1-3
A08 01  1  ENG  @1 Smooth words on 2-letter alphabets having same parity
A11 01  1    @1 BRLEK (S.)
A11 02  1    @1 JAMET (D.)
A11 03  1    @1 PAQUIN (G.)
A14 01      @1 LaCIM, Université du Québec à Montréal, C.P 8888 Succursale "Centre-Ville" @2 Montréal (QC), H3C 3P8 @3 CAN @Z 1 aut. @Z 3 aut.
A14 02      @1 LORIA -Campus Scientifique -BP 239 @2 54506, Vandoeuvre-lès-Nancy @3 FRA @Z 2 aut.
A20       @1 166-181
A21       @1 2008
A23 01      @0 ENG
A43 01      @1 INIST @2 17243 @5 354000183747600140
A44       @0 0000 @1 © 2008 INIST-CNRS. All rights reserved.
A45       @0 16 ref.
A47 01  1    @0 08-0195807
A60       @1 P
A61       @0 A
A64 01  1    @0 Theoretical computer science
A66 01      @0 NLD
C01 01    ENG  @0 In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b are odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b-1+1).
C02 01  X    @0 001D02A08
C02 02  X    @0 001D02A05
C03 01  X  FRE  @0 Mot infini @5 17
C03 01  X  ENG  @0 Infinite word @5 17
C03 01  X  SPA  @0 Palabra infinita @5 17
C03 02  X  FRE  @0 Lettre alphabet @5 18
C03 02  X  ENG  @0 Letter @5 18
C03 02  X  SPA  @0 Letra alfabeto @5 18
C03 03  X  FRE  @0 Alphabet @5 19
C03 03  X  ENG  @0 Alphabet @5 19
C03 03  X  SPA  @0 Alfabeto @5 19
C03 04  X  FRE  @0 Parité @5 20
C03 04  X  ENG  @0 Parity @5 20
C03 04  X  SPA  @0 Paridad @5 20
C03 05  X  FRE  @0 Nombre entier @5 21
C03 05  X  ENG  @0 Integer @5 21
C03 05  X  SPA  @0 Entero @5 21
C03 06  X  FRE  @0 Fermeture @5 22
C03 06  X  ENG  @0 Closure @5 22
C03 06  X  SPA  @0 Cerradura @5 22
C03 07  X  FRE  @0 Temps linéaire @5 23
C03 07  X  ENG  @0 Linear time @5 23
C03 07  X  SPA  @0 Tiempo lineal @5 23
C03 08  X  FRE  @0 Calcul automatique @5 24
C03 08  X  ENG  @0 Computing @5 24
C03 08  X  SPA  @0 Cálculo automático @5 24
C03 09  X  FRE  @0 Ordre lexicographique @5 25
C03 09  X  ENG  @0 Lexicographic order @5 25
C03 09  X  SPA  @0 Orden lexicográfico @5 25
C03 10  X  FRE  @0 Raccordement @5 26
C03 10  X  ENG  @0 Connection @5 26
C03 10  X  SPA  @0 Conexión @5 26
C03 11  X  FRE  @0 Fréquence @5 27
C03 11  X  ENG  @0 Frequency @5 27
C03 11  X  SPA  @0 Frecuencia @5 27
C03 12  X  FRE  @0 Factorisation @5 28
C03 12  X  ENG  @0 Factorization @5 28
C03 12  X  SPA  @0 Factorización @5 28
C03 13  X  FRE  @0 Informatique théorique @5 29
C03 13  X  ENG  @0 Computer theory @5 29
C03 13  X  SPA  @0 Informática teórica @5 29
C03 14  X  FRE  @0 Algorithme linéaire @4 INC @5 70
C03 15  X  FRE  @0 Algorithme temps linéaire @4 INC @5 71
C03 16  X  FRE  @0 68Wxx @4 INC @5 72
C03 17  X  FRE  @0 Mot Lyndon @4 INC @5 73
N21       @1 119
N44 01      @1 OTO
N82       @1 OTO

Format Inist (serveur)

NO : PASCAL 08-0195807 INIST
ET : Smooth words on 2-letter alphabets having same parity
AU : BRLEK (S.); JAMET (D.); PAQUIN (G.)
AF : LaCIM, Université du Québec à Montréal, C.P 8888 Succursale "Centre-Ville"/Montréal (QC), H3C 3P8/Canada (1 aut., 3 aut.); LORIA -Campus Scientifique -BP 239/54506, Vandoeuvre-lès-Nancy/France (2 aut.)
DT : Publication en série; Niveau analytique
SO : Theoretical computer science; ISSN 0304-3975; Coden TCSCDI; Pays-Bas; Da. 2008; Vol. 393; No. 1-3; Pp. 166-181; Bibl. 16 ref.
LA : Anglais
EA : In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b are odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b-1+1).
CC : 001D02A08; 001D02A05
FD : Mot infini; Lettre alphabet; Alphabet; Parité; Nombre entier; Fermeture; Temps linéaire; Calcul automatique; Ordre lexicographique; Raccordement; Fréquence; Factorisation; Informatique théorique; Algorithme linéaire; Algorithme temps linéaire; 68Wxx; Mot Lyndon
ED : Infinite word; Letter; Alphabet; Parity; Integer; Closure; Linear time; Computing; Lexicographic order; Connection; Frequency; Factorization; Computer theory
SD : Palabra infinita; Letra alfabeto; Alfabeto; Paridad; Entero; Cerradura; Tiempo lineal; Cálculo automático; Orden lexicográfico; Conexión; Frecuencia; Factorización; Informática teórica
LO : INIST-17243.354000183747600140
ID : 08-0195807

Links to Exploration step

Pascal:08-0195807

Le document en format XML

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<s5>28</s5>
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<s0>Factorization</s0>
<s5>28</s5>
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<s0>Factorización</s0>
<s5>28</s5>
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<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="ENG">
<s0>Computer theory</s0>
<s5>29</s5>
</fC03>
<fC03 i1="13" i2="X" l="SPA">
<s0>Informática teórica</s0>
<s5>29</s5>
</fC03>
<fC03 i1="14" i2="X" l="FRE">
<s0>Algorithme linéaire</s0>
<s4>INC</s4>
<s5>70</s5>
</fC03>
<fC03 i1="15" i2="X" l="FRE">
<s0>Algorithme temps linéaire</s0>
<s4>INC</s4>
<s5>71</s5>
</fC03>
<fC03 i1="16" i2="X" l="FRE">
<s0>68Wxx</s0>
<s4>INC</s4>
<s5>72</s5>
</fC03>
<fC03 i1="17" i2="X" l="FRE">
<s0>Mot Lyndon</s0>
<s4>INC</s4>
<s5>73</s5>
</fC03>
<fN21>
<s1>119</s1>
</fN21>
<fN44 i1="01">
<s1>OTO</s1>
</fN44>
<fN82>
<s1>OTO</s1>
</fN82>
</pA>
</standard>
<server>
<NO>PASCAL 08-0195807 INIST</NO>
<ET>Smooth words on 2-letter alphabets having same parity</ET>
<AU>BRLEK (S.); JAMET (D.); PAQUIN (G.)</AU>
<AF>LaCIM, Université du Québec à Montréal, C.P 8888 Succursale "Centre-Ville"/Montréal (QC), H3C 3P8/Canada (1 aut., 3 aut.); LORIA -Campus Scientifique -BP 239/54506, Vandoeuvre-lès-Nancy/France (2 aut.)</AF>
<DT>Publication en série; Niveau analytique</DT>
<SO>Theoretical computer science; ISSN 0304-3975; Coden TCSCDI; Pays-Bas; Da. 2008; Vol. 393; No. 1-3; Pp. 166-181; Bibl. 16 ref.</SO>
<LA>Anglais</LA>
<EA>In this paper, we consider smooth words over 2-letter alphabets {a, b}, where a, b are integers having same parity, with 0 < a < b. We show that all are recurrent and that the closure of the set of factors under reversal holds for odd alphabets only. We provide a linear time algorithm computing the extremal words, w.r.t. lexicographic order. The minimal word is an infinite Lyndon word if and only if either a = 1 and b are odd, or a, b are even. A connection is established between generalized Kolakoski words and maximal infinite smooth words over even 2-letter alphabets revealing new properties for some of the generalized Kolakoski words. Finally, the frequency of letters in extremal words is 1/2 for even alphabets, and for a = 1 with b odd, the frequency of b's is 1/(√2b-1+1).</EA>
<CC>001D02A08; 001D02A05</CC>
<FD>Mot infini; Lettre alphabet; Alphabet; Parité; Nombre entier; Fermeture; Temps linéaire; Calcul automatique; Ordre lexicographique; Raccordement; Fréquence; Factorisation; Informatique théorique; Algorithme linéaire; Algorithme temps linéaire; 68Wxx; Mot Lyndon</FD>
<ED>Infinite word; Letter; Alphabet; Parity; Integer; Closure; Linear time; Computing; Lexicographic order; Connection; Frequency; Factorization; Computer theory</ED>
<SD>Palabra infinita; Letra alfabeto; Alfabeto; Paridad; Entero; Cerradura; Tiempo lineal; Cálculo automático; Orden lexicográfico; Conexión; Frecuencia; Factorización; Informática teórica</SD>
<LO>INIST-17243.354000183747600140</LO>
<ID>08-0195807</ID>
</server>
</inist>
</record>

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