Fraenkel word: Difference between revisions
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=== Other conjectures === | === Other conjectures === | ||
'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then {MV(''F''<sub>2''k''−1</sub>), MV(''F''<sub>2''k''</sub>), MV(''F''<sub>2''k''+1</sub>)} is an arithmetic progression with common difference ''f''<sub>2''k''</sub> (the 2''k''-th Fibonacci number: 1, 3, 8, 21, ...) for every ''k'' ≥ 1. | '''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then {MV(''F''<sub>2''k''−1</sub>), MV(''F''<sub>2''k''</sub>), MV(''F''<sub>2''k''+1</sub>)} is an arithmetic progression with common difference ''f''<sub>2''k''</sub> (the 2''k''-th Fibonacci number: 1, 3, 8, 21, ...) for every ''k'' ≥ 1. | ||
'''Conjecture:''' For all ''k'' > 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 ''k'' > 1, MV(''G<sub>k</sub>'') = 3×2<sup>k-1</sup>-1. | |||
'''Conjecture:''' For all ''k'' > 1, MV(''G<sub>k</sub>'''n''''') = 2<sup>k</sup>. | |||
== See also == | == See also == | ||