Balanced word: Difference between revisions
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A | A scale is '''balanced''' if it satisfies a certain (quite strong) restriction on how much the intervals in within any of the scale's interval classes can differ. | ||
<math> \mathsf{block\_balance}(s) := \max \big\{ \big| |w|_{a} - |w'|_{a} \big| : | Mathematically, let ''a'' be a letter in a [[word]] or [[necklace]] ''s''. Define | ||
<math> \mathsf{block\_balance}(s, a) := \max \big\{ \big| |w|_{a} - |w'|_{a} \big| : w, w'\text{ are length-}k \text{ subwords of } s\big\},</math> | |||
where |''u''|<sub>''a''</sub> is the number of occurrences of the letter ''a'' in the word ''u''. | where |''u''|<sub>''a''</sub> is the number of occurrences of the letter ''a'' in the word ''u''. | ||
Then ''s'' is '''balanced''' if its '''block balance'''{{idiosyncratic}} satisfies the following: | |||
<math> \mathsf{block\_balance}(s) := \max \big\{ \mathsf{block\_balance}(s, a) : a \text{ is a letter of }s \big\} \leq 1,</math> | |||
== Properties == | == Properties == | ||