Hypercubic billiard word: Difference between revisions
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* There exist ternary billiard scales of length up to 29 that do not have a well-formed [[generator sequence]]. | * There exist ternary billiard scales of length up to 29 that do not have a well-formed [[generator sequence]]. | ||
* A ''d''-ary billiard scale is (''d'' − 1)-[[balance|block balanced]] and (''d'' − 1)-[[balance|chain balanced]].<ref name="vuillon"/><ref name="sano">Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> However, scales on at least 3 letters satisfying the block balance bound need not be billiard scales. | * A ''d''-ary billiard scale is (''d'' − 1)-[[balance|block balanced]] and (''d'' − 1)-[[balance|chain balanced]].<ref name="vuillon"/><ref name="sano">Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> However, scales on at least 3 letters satisfying the block balance bound need not be billiard scales. | ||
* An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> | * An aperiodic ''d''-ary billiard scale has maximum variety at most 2<sup>''d''−1</sup>.<ref name="andrieu">Mélodie Andrieu, Léo Vivion. Minimal Complexities for Infinite Words Written with d Letters. Combinatorics on Words - 14th International Conference, WORDS, Jun 2023, Umeå, Sweden. pp.3–13.</ref> | ||
* If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4. | * If a ternary billiard scale has a well-formed generator sequence, the WFGS must use either two or three distinct generators, since ternary billiard scales are MV3 or MV4. | ||
* A ''d''-ary billiard scale satisfies the two generalizations of balance listed on the article [[balanced word]]. | * A ''d''-ary billiard scale satisfies the two generalizations of balance listed on the article [[balanced word]]. | ||