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.

* Not all billiard words of arity higher than 2 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>

* All ternary billiard scales appear to have [[balance]] at most 2. However, ternary balance-2 scales need not be billiard scales.

* All ternary billiard scales appear to have maximum variety at most 4 and [[balance]] at most 2. However, ternary balance-2 scales need not be billiard scales.

=== Determining whether a scale word is a billiard scale ===

@@ Line 16: / Line 16: @@
 ** That’s because projecting we just remove some of the αs from the list, leaving all remaining ones intact.
 * Not all billiard words of arity higher than 2 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>
-* All ternary billiard scales appear to have [[balance]] at most 2. However, ternary balance-2 scales need not be billiard scales.
+* All ternary billiard scales appear to have maximum variety at most 4 and [[balance]] at most 2. However, ternary balance-2 scales need not be billiard scales.
 === Determining whether a scale word is a billiard scale ===