Fraenkel word: Difference between revisions

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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''. Then {MV(''F''2''k''−1), MV(''F''2''k''), MV(''F''2''k''+1)} is an arithmetic progression with common difference ''f''''k''+1 (the (''k'' ~~+ 1)~~-st Fibonacci number) for every ''k'' ≥ 1.

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

== See also ==

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 For [[circular word]]s (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.
 === Other conjectures ===
-'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then {MV(''F''<sub>2''k''&minus;1</sub>), MV(''F''<sub>2''k''</sub>), MV(''F''<sub>2''k''+1</sub>)} is an arithmetic progression with common difference ''f''<sub>''k''+1</sub> (the (''k'' + 1)-st Fibonacci number) for every ''k'' &ge; 1.
+'''Conjecture:''' Let MV(''s'') denote the [[maximum variety]] of the circular word ''s''. Then {MV(''F''<sub>2''k''&minus;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'' &ge; 1.
 == See also ==