Fraenkel word: Difference between revisions

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A '''Fraenkel word''' over ''n'' letters is defined recursively by

In combinatorics on words, the '''Fraenkel word''' over ''n'' letters <math>\mathbf{0}, \mathbf{1}, ..., (\mathbf{n-1})</math> is defined recursively by

<math>\displaystyle{

\begin{align*}

F_0 &= \epsilon, \\

F_1 &= \mathbf{0}, \\

F_2 &= \mathbf{010}, \\

F_3 &= \mathbf{0102010}, \\

&\ \ \vdots \\

F_{n} &= F_{n-1}(\mathbf{n-1})F_{n-1}.

F_{n} &= F_{n-1}(\mathbf{n-1})F_{n-1},

\end{align*}}

</math>

where ε is the empty word. Fraenkel words may be encountered as exceptional examples of scale properties alongside larger families, e.g. for the [[interval variety|maximum/strict variety]] 3 and [[balance]] properties.

Fraenkel words are named after mathematician Aviezri S. Fraenkel.

== Facts ==

Below we denote the length of a word ''w'' by {{abs|''w''}} and the number of occurrences of the letter '''i''' in ''w'' as {{abs|''w''}}'''i''', as is standard notation in combinatorics on words. The notation ''w''(''u''0, ..., {{nowrap|''u''''r'' − 1}}) represents the word ''w'' in '''0''', '''1''', ..., {{nowrap|'''''r'' − 1'''}} but with '''i''' replaced by the word ''u''''i''.

=== Fraenkel words are balanced ===

{{theorem|contents=As circular words, Fraenkel words are [[balanced]].}}

The theorem will be a consequence of the following lemmas:

{{theorem|name=Lemma|contents=Let ''F''''n'' denote the non-circular Fraenkel word on ''n'' letters. For {{nowrap|''n'' ≥ 1}} and {{nowrap|0 ≤ ''i'' ≤ ''n'' − 1}}, the letter '''i''' appears once every {{nowrap|2''i'' + 1}} letters in ''F''''n''; i.e. in every subword of the form '''i'''''w'''''i''', {{nowrap|{{abs|''w''}} {{=}} 2''i'' + 1 − 1}}.}}

{{proof|contents=We prove this by induction on ''n''. The {{nowrap|''n'' {{=}} 1}} case being trivial, for {{nowrap|''n'' > 1}} we start by observing that '''i''' always occurs as the greatest letter of a subword that is the Fraenkel word {{nowrap|''F''''i'' + 1}}, and {{nowrap|''F''''i'' + 1}} is always flanked (on at least one side) by letters that are greater. This subword '''i'''''F''''i'' is not a suffix of ''F''''n'', and we thus have

'''i'''''F''''i'''''k'''...

where {{nowrap|''k'' > ''i''}}, and ''F''0 is the empty word. Since '''k''' occurs as the middle letter of {{nowrap|''F''''k'' + 1}}, there is a copy of ''F''''k'' that follows '''k'''; ''F''''k'' has ''F''i as a prefix. Thus ''F''''n'' has a subword

'''i'''''F''''i'''''k'''''F''''i'''''i''',

as desired, since {{nowrap|{{abs|''F''i}} {{=}} 2''i'' − 1}}.

}}

{{theorem|name=Lemma|contents=For all {{nowrap|''n'' ≥ 1|0 ≤ ''i'' ≤ ''n'' − 1}}, and {{nowrap|1 ≤ {{abs|''w''}} ≤ 2''n''/2 − 2}}, the following holds for any subword ''w'' of ''F''''n'':

# If {{nowrap|{{abs|''w''}} ≡ 0 (mod 2''i'' + 1)}}, then {{nowrap|{{abs|''w''}}'''i''' {{=}} {{abs|''w''}}/2''i'' + 1}}.

# If {{nowrap|{{abs|''w''}} ≢ 0 (mod 2''i'' + 1)}}, then {{abs|''w''}}'''i''' {{=}} either {{nowrap|{{floor|{{abs|''w''}}/2''i'' + 1|1.75}}}} or {{nowrap|{{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}}}.

#* More precisely, if for a given ''i'' we have {{nowrap|''w'' {{=}} ''uv''}} or ''vu'' where ''u'' is a possibly empty word whose length is {{nowrap|0 (mod 2''i'' + 1)}}, and ''v'' is a nonempty word intersecting the middle of an {{nowrap|''F''''i'' + 1}}, then {{nowrap|{{abs|''w''}}'''i''' {{=}} {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}}}. Otherwise, {{nowrap|{{abs|''w''}}'''i''' {{=}} {{floor|{{abs|''w''}}/2''i'' + 1|1.75}}}}.

}}

{{proof|contents=We use the previous lemma. In the first case, ''w'' is guaranteed to have exactly ''k''-many '''i'''s where {{abs|''w''}} = ''k''2''i'' + 1. In the second case, if ''k''2''i'' + 1 < {{abs|''w''}} < (''k'' + 1)2''i'' + 1 and ''w'' = ''uv'' or ''vu'' where {{abs|''u''}} ≡ 0 mod 2''i'' + 1, then ''u'' satisfies {{abs|''u''}}'''i''' = ''k''2''i'' + 1/2''i'' + 1 = ''k'' by the previous case. Thus {{abs|''w''}}'''i''' is determined by {{abs|''v''}}'''i''', which is 1 if ''v'' contains the '''i''' in the middle of ''F''''i'', implying {{abs|''w''}}'''i''' = {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}, and 0 otherwise, implying {{abs|''w''}}'''i''' = floor({{abs|''w''}}/2''i'' + 1).}}

{{theorem|name=Lemma|contents=Let [''F''''n''] denote the circular Fraenkel word on ''n'' letters. Suppose ''w'' is a proper subword of [''F''''n''] such that {{nowrap|''w'' {{=}} ''uv''}} where ''u'' is a nonempty suffix of ''F''''n'' and ''v'' is a nonempty prefix of ''F''''n''. For {{nowrap|1 ≤ {{abs|''w''}} ≤ 2''n''/2 − 2}}, either {{nowrap|{{abs|''w''}}'''i''' {{=}} {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}}} or {{nowrap|{{ceil|{{abs|''w''}}/2''i'' + 1|1.75}} − 1}}.

}}

{{proof|contents=

There are 2 cases:

# Both {{abs|''u''}} and {{abs|''v''}} are {{nowrap|0 (mod 2''i'' + 1)}}.

# At least one of {{abs|''u''}} and {{abs|''v''}} is not {{nowrap|0 (mod 2''i'' + 1)}}.

In case 1, by the preceding lemma {{nowrap|{{abs|''u''}}'''i''' {{=}} {{abs|''u''}}/2''i'' + 1}} and {{nowrap|{{abs|''v''}}'''i''' {{=}} {{abs|''v''}}/2''i'' + 1}}, and hence {{nowrap|{{abs|''w''}}'''i''' {{=}} {{abs|''w''}}/2''i'' + 1}} {{nowrap|{{=}} {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}}}.

In case 2, suppose {{nowrap|''w'' {{=}} ''ustv''}} where ''st'' is as in case 1 and {{abs|''u''}} and {{abs|''v''}} are less than {{nowrap|2''i'' + 1}}. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''''i''. Using {{nowrap|{{abs|''st''}} {{=}} {{abs|''st''}}'''i'''2''i'' + 1}}, we have

{{nowrap|{{abs|''st''}}'''i'''2''i'' + 1 < {{abs|''w''}}}} {{nowrap|{{=}} {{abs|''st''}}'''i'''2''i'' + 1 + {{abs|''u''}} + {{abs|''v''}}}} {{nowrap|≤ {{abs|''st''}}'''i'''2''i'' + 1 + 2''i'' + 1 − 2}},

thus

{{nowrap|{{abs|''w''}}'''i''' ≥ {{abs|''st''}}'''i'''}} {{nowrap|{{=}} {{abs|''st''}}/2''i'' + 1}} {{nowrap|{{=}} {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}} − 1}}.

On the other hand, {{nowrap|{{abs|''w''}}'''i''' < {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}}}}, lest ''u'' or ''v'' have an '''i'''. Therefore {{nowrap|{{abs|''w''}}'''i''' {{=}} {{ceil|{{abs|''w''}}/2''i'' + 1|1.75}} − 1}}.

}}

== Open problems ==

For circular ~~words~~ (equivalently, infinite periodic words), '''Fraenkel's conjecture''' asserts that the only [[balanced]] circular words over ''n'' ≥ 3 letters with letter occurrences pairwise distinct are (letter reassignments of) <math>F_n.</math><ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> The conjecture is known to be true for 3 ≤ ''n'' ≤ 7.

=== Fraenkel's conjecture ===

For [[circular word]]s (equivalently, infinite periodic words), '''Fraenkel's conjecture''' asserts that the only [[balanced]] circular words over {{nowrap|''n'' ≥ 3}} letters with letter occurrences pairwise distinct are (letter reassignments of) <math>F_n.</math><ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> The conjecture is known to be true for {{nowrap|3 ≤ ''n'' ≤ 7}}.

=== Other conjectures ===

'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then <nowiki>{</nowiki>{{nowrap|MV(''F''2''k'' − 1)}}, MV(''F''2''k''), {{nowrap|MV(''F''2''k'' + 1)}}<nowiki>}</nowiki> is an arithmetic progression with common difference ''f''2''k'' (the 2''k''-th Fibonacci number: 1, 3, 8, 21, ...) for every {{nowrap|''k'' ≥ 1}}.

'''Conjecture:''' For all {{nowrap|''k'' > 0|MV(''Fk'''n''''') {{=}} ''f''''k'' + 1}}.

Let Gk be a modified Fraenkel word, defined by

<math>\displaystyle{

\begin{align*}

G_0 &= \epsilon, \\

G_1 &= \mathbf{0}, \\

G_2 &= \mathbf{01010}, \\

G_3 &= \mathbf{01010201010201010}, \\

&\ \ \vdots \\

G_{n} &= G_{n-1}(\mathbf{n-1})G_{n-1}(\mathbf{n-1})G_{n-1}.

\end{align*}}

</math>

'''Conjecture:''' For all {{nowrap|''k'' > 1}}, {{nowrap|MV(''Gk'') {{=}} 3{{dot}}2''k'' − 1 − 1}}.

'''Conjecture:''' For all {{nowrap|''k'' > 1}}, {{nowrap|MV(''Gk'''n''''') {{=}} 2''k''}}.

== See also ==

* [[ABACABA JI scales]]

* [[ABACABADABACABA JI scales]]

== References ==

[[Category:Terms]]

[[Category:Combinatorics on words]]

[[Category:Math]]

[[Category:~~Articles~~ with open problems]]

[[Category:Pages with open problems]]

@@ Line 1: / Line 1: @@
-A '''Fraenkel word''' over ''n'' letters is defined recursively by
+In combinatorics on words, the '''Fraenkel word''' over ''n'' letters <math>\mathbf{0}, \mathbf{1}, ..., (\mathbf{n-1})</math> is defined recursively by
 <math>\displaystyle{
 \begin{align*}
+F_0 &= \epsilon, \\
 F_1 &= \mathbf{0}, \\
 F_2 &= \mathbf{010}, \\
 F_3 &= \mathbf{0102010}, \\
 &\ \ \vdots \\
-F_{n} &= F_{n-1}(\mathbf{n-1})F_{n-1}.
+F_{n} &= F_{n-1}(\mathbf{n-1})F_{n-1},
 \end{align*}}
 </math>
+where ε is the empty word. Fraenkel words may be encountered as exceptional examples of scale properties alongside larger families, e.g. for the [[interval variety|maximum/strict variety]] 3 and [[balance]] properties.
+Fraenkel words are named after mathematician Aviezri S. Fraenkel.
+== Facts ==
+Below we denote the length of a word ''w'' by {{abs|''w''}} and the number of occurrences of the letter '''i''' in ''w'' as {{abs|''w''}}<sub>'''i'''</sub>, as is standard notation in combinatorics on words. The notation ''w''(''u''<sub>0</sub>, ..., {{nowrap|''u''<sub>''r'' − 1</sub>}}) represents the word ''w'' in '''0''', '''1''', ..., {{nowrap|'''''r'' − 1'''}} but with '''i''' replaced by the word ''u''<sub>''i''</sub>.
+=== Fraenkel words are balanced ===
+{{theorem|contents=As circular words, Fraenkel words are [[balanced]].}}
+The theorem will be a consequence of the following lemmas:
+{{theorem|name=Lemma|contents=Let ''F''<sub>''n''</sub> denote the non-circular Fraenkel word on ''n'' letters. For {{nowrap|''n'' &ge; 1}} and {{nowrap|0 &le; ''i'' &le; ''n'' − 1}}, the letter '''i''' appears once every {{nowrap|2<sup>''i'' + 1</sup>}} letters in ''F''<sub>''n''</sub>; i.e. in every subword of the form '''i'''''w'''''i''', {{nowrap|{{abs|''w''}} {{=}} 2<sup>''i'' + 1</sup> − 1}}.}}
+{{proof|contents=We prove this by induction on ''n''. The {{nowrap|''n'' {{=}} 1}} case being trivial, for {{nowrap|''n'' &gt; 1}} we start by observing that '''i''' always occurs as the greatest letter of a subword that is the Fraenkel word {{nowrap|''F''<sub>''i'' + 1</sub>}}, and {{nowrap|''F''<sub>''i'' + 1</sub>}} is always flanked (on at least one side) by letters that are greater. This subword '''i'''''F''<sub>''i''</sub> is not a suffix of ''F''<sub>''n''</sub>, and we thus have
+'''i'''''F''<sub>''i''</sub>'''k'''...
+where {{nowrap|''k'' &gt; ''i''}}, and ''F''<sub>0</sub> is the empty word. Since '''k''' occurs as the middle letter of {{nowrap|''F''<sub>''k'' + 1</sub>}}, there is a copy of ''F''<sub>''k''</sub> that follows '''k'''; ''F''<sub>''k''</sub> has ''F''<sub>i</sub> as a prefix. Thus ''F''<sub>''n''</sub> has a subword
+'''i'''''F''<sub>''i''</sub>'''k'''''F''<sub>''i''</sub>'''i''',
+as desired, since {{nowrap|{{abs|''F''<sub>i</sub>}} {{=}} 2<sup>''i''</sup> − 1}}.
+}}
+{{theorem|name=Lemma|contents=For all {{nowrap|''n'' &ge; 1|0 &le; ''i'' &le; ''n'' − 1}}, and {{nowrap|1 &le; {{abs|''w''}} &le; 2<sup>''n''/2</sup> − 2}}, the following holds for any subword ''w'' of ''F''<sub>''n''</sub>:
+# If {{nowrap|{{abs|''w''}} ≡ 0 (mod 2<sup>''i'' + 1</sup>)}}, then {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{abs|''w''}}/2<sup>''i'' + 1</sup>}}.
+# If {{nowrap|{{abs|''w''}} ≢ 0 (mod 2<sup>''i'' + 1</sup>)}}, then {{abs|''w''}}<sub>'''i'''</sub> {{=}} either {{nowrap|{{floor|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}} or {{nowrap|{{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}}.
+#* More precisely, if for a given ''i'' we have {{nowrap|''w'' {{=}} ''uv''}} or ''vu'' where ''u'' is a possibly empty word whose length is {{nowrap|0 (mod 2<sup>''i'' + 1</sup>)}}, and ''v'' is a nonempty word intersecting the middle of an {{nowrap|''F''<sub>''i'' + 1</sub>}}, then {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}}. Otherwise, {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{floor|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}}.
+}}
+{{proof|contents=We use the previous lemma. In the first case, ''w'' is guaranteed to have exactly ''k''-many '''i'''s where {{abs|''w''}} = ''k''2<sup>''i'' + 1</sup>. In the second case, if ''k''2<sup>''i'' + 1</sup> < {{abs|''w''}} < (''k'' + 1)2<sup>''i'' + 1</sup> and ''w'' = ''uv'' or ''vu'' where {{abs|''u''}} ≡ 0 mod 2<sup>''i'' + 1</sup>,  then ''u'' satisfies {{abs|''u''}}<sub>'''i'''</sub> = ''k''2<sup>''i'' + 1</sup>/2<sup>''i'' + 1</sup> = ''k'' by the previous case. Thus {{abs|''w''}}<sub>'''i'''</sub>  is determined by {{abs|''v''}}<sub>'''i'''</sub>, which is 1 if ''v'' contains the '''i''' in the middle of ''F''<sub>''i''</sub>, implying {{abs|''w''}}<sub>'''i'''</sub> = {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}, and 0 otherwise, implying {{abs|''w''}}<sub>'''i'''</sub> = floor({{abs|''w''}}/2<sup>''i'' + 1</sup>).}}
+{{theorem|name=Lemma|contents=Let [''F''<sub>''n''</sub>] denote the circular Fraenkel word on ''n'' letters. Suppose ''w'' is a proper subword of [''F''<sub>''n''</sub>] such that {{nowrap|''w'' {{=}} ''uv''}} where ''u'' is a nonempty suffix of ''F''<sub>''n''</sub> and ''v'' is a nonempty prefix of ''F''<sub>''n''</sub>. For {{nowrap|1 &le; {{abs|''w''}} &le; 2<sup>''n''/2</sup> − 2}}, either {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}} or {{nowrap|{{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}} − 1}}.
+}}
+{{proof|contents=
+There are 2 cases:
+# Both {{abs|''u''}} and {{abs|''v''}} are {{nowrap|0 (mod 2<sup>''i'' + 1</sup>)}}.
+# At least one of {{abs|''u''}} and {{abs|''v''}} is not {{nowrap|0 (mod 2<sup>''i'' + 1</sup>)}}.
+In case 1, by the preceding lemma {{nowrap|{{abs|''u''}}<sub>'''i'''</sub> {{=}} {{abs|''u''}}/2<sup>''i'' + 1</sup>}} and {{nowrap|{{abs|''v''}}<sub>'''i'''</sub> {{=}} {{abs|''v''}}/2<sup>''i'' + 1</sup>}}, and hence {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{abs|''w''}}/2<sup>''i'' + 1</sup>}} {{nowrap|{{=}} {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}}.
+In case 2, suppose {{nowrap|''w'' {{=}} ''ustv''}} where ''st'' is as in case 1 and {{abs|''u''}} and {{abs|''v''}} are less than {{nowrap|2<sup>''i'' + 1</sup>}}. Neither ''u'' nor ''v'' can contain an '''i''', as they are subwords of ''F''<sub>''i''</sub>. Using {{nowrap|{{abs|''st''}} {{=}} {{abs|''st''}}<sub>'''i'''</sub>2<sup>''i'' + 1</sup>}}, we have
+{{nowrap|{{abs|''st''}}<sub>'''i'''</sub>2<sup>''i'' + 1</sup> &lt; {{abs|''w''}}}} {{nowrap|{{=}} {{abs|''st''}}<sub>'''i'''</sub>2<sup>''i'' + 1</sup> + {{abs|''u''}} + {{abs|''v''}}}} {{nowrap|&le; {{abs|''st''}}<sub>'''i'''</sub>2<sup>''i'' + 1</sup> + 2<sup>''i'' + 1</sup> − 2}},
+thus
+{{nowrap|{{abs|''w''}}<sub>'''i'''</sub> &ge; {{abs|''st''}}<sub>'''i'''</sub>}} {{nowrap|{{=}} {{abs|''st''}}/2<sup>''i'' + 1</sup>}} {{nowrap|{{=}} {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}} − 1}}.
+On the other hand, {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> &lt; {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}}}}, lest ''u'' or ''v'' have an '''i'''. Therefore {{nowrap|{{abs|''w''}}<sub>'''i'''</sub> {{=}} {{ceil|{{abs|''w''}}/2<sup>''i'' + 1</sup>|1.75}} − 1}}.
+}}
 == Open problems ==
-For circular words (equivalently, infinite periodic words), '''Fraenkel's conjecture''' asserts that the only [[balanced]] circular words over ''n'' &ge; 3 letters with letter occurrences pairwise distinct are (letter reassignments of) <math>F_n.</math><ref>Bulgakova, D. V., Buzhinsky, N., & Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> The conjecture is known to be true for 3 &le; ''n'' &le; 7.
+=== Fraenkel's conjecture ===
+For [[circular word]]s (equivalently, infinite periodic words), '''Fraenkel's conjecture''' asserts that the only [[balanced]] circular words over {{nowrap|''n'' &ge; 3}} letters with letter occurrences pairwise distinct are (letter reassignments of) <math>F_n.</math><ref>Bulgakova, D. V., Buzhinsky, N., &amp; Goncharov, Y. O. (2023). On balanced and abelian properties of circular words over a ternary alphabet. Theoretical Computer Science, 939, 227-236.</ref> The conjecture is known to be true for {{nowrap|3 &le; ''n'' &le; 7}}.
+=== Other conjectures ===
+'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then <nowiki>{</nowiki>{{nowrap|MV(''F''<sub>2''k'' − 1</sub>)}}, MV(''F''<sub>2''k''</sub>), {{nowrap|MV(''F''<sub>2''k'' + 1</sub>)}}<nowiki>}</nowiki> is an arithmetic progression with common difference ''f''<sub>2''k''</sub> (the 2''k''-th Fibonacci number: 1, 3, 8, 21, ...) for every {{nowrap|''k'' &ge; 1}}.
+'''Conjecture:''' For all {{nowrap|''k'' &gt; 0|MV(''F<sub>k</sub>'''n''''') {{=}} ''f''<sub>''k'' + 1</sub>}}.
+Let G<sub>k</sub> be a modified Fraenkel word, defined by
+<math>\displaystyle{
+\begin{align*}
+G_0 &= \epsilon, \\
+G_1 &= \mathbf{0}, \\
+G_2 &= \mathbf{01010}, \\
+G_3 &= \mathbf{01010201010201010}, \\
+&\ \ \vdots \\
+G_{n} &= G_{n-1}(\mathbf{n-1})G_{n-1}(\mathbf{n-1})G_{n-1}.
+\end{align*}}
+</math>
+'''Conjecture:''' For all {{nowrap|''k'' > 1}}, {{nowrap|MV(''G<sub>k</sub>'') {{=}} 3{{dot}}2<sup>''k'' − 1</sup> − 1}}.
+'''Conjecture:''' For all {{nowrap|''k'' > 1}}, {{nowrap|MV(''G<sub>k</sub>'''n''''') {{=}} 2<sup>''k''</sup>}}.
+== See also ==
+* [[ABACABA JI scales]]
+* [[ABACABADABACABA JI scales]]
 == References ==
+<references />
 [[Category:Terms]]
 [[Category:Combinatorics on words]]
 [[Category:Math]]
-[[Category:Articles with open problems]]
+[[Category:Pages with open problems]]