Fraenkel word: Difference between revisions
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== Open problems == | == Open problems == | ||
=== Fraenkel's conjecture === | |||
For [[circular word]]s (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. | For [[circular word]]s (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. | ||
=== Other conjectures === | |||
'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. For every ''k'' ≥ 1, MV(''F''<sub>2''k''</sub>) − MV(''F''<sub>2''k''−1</sub>) = MV(''F''<sub>2''k''+1</sub>) − MV(''F''<sub>2''k''</sub>). | |||
== See also == | == See also == | ||
* [[ABACABA JI scales]] | * [[ABACABA JI scales]] | ||