Hypercubic billiard word: Difference between revisions
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** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact. | ** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact. | ||
* Not all billiard words over more than 2 letters are [[balance|balanced]]. (In binary scales, the term "(1-)balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are 1-balanced are [[MV3]].<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> | * Not all billiard words over more than 2 letters are [[balance|balanced]]. (In binary scales, the term "(1-)balanced" becomes one characterization of the [[MOS]] property.) Ternary billiard scales that are 1-balanced are [[MV3]].<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> | ||
* | * Balanced ternary billiard scales with odd length are pairwise well-formed and have a well-formed [[generator sequence]] of two terms; see [[Ternary scale theorems]]. | ||
* 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 has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying this bound need not be billiard scales. | * A ''d''-ary billiard scale has [[balance]] at most (''d'' − 1).<ref name="vuillon"/> However, scales on at least 3 letters satisfying this bound need not be billiard scales. | ||