Balanced word: Difference between revisions

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== Generalizations ==

# If <math> \mathsf{block\_balance}(s) \leq m,</math> then we say that ''s'' is ''m''-'''block-balanced'''{{idiosyncratic}}.

# The following stronger property implies ''m''-block-balancedness for ''m'' ≥ 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(w') = len(''w'') + ''m'' + 1 satisfies |w'|''a'' ≥ |''w''|''a'' + 1.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word ~~is a proof~~ that for binary scales, balanced implies 1-chain-balanced.

# The following stronger property implies ''m''-block-balancedness for ''m'' ≥ 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(w') = len(''w'') + ''m'' + 1 satisfies |w'|''a'' ≥ |''w''|''a'' + 1.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)

== References ==

[[Category:Scale]][[Category:Terms]]

[[Category:Combinatorics on words]]

@@ Line 15: / Line 15: @@
 == Generalizations ==
 # If <math> \mathsf{block\_balance}(s) \leq m,</math> then we say that ''s'' is ''m''-'''block-balanced'''{{idiosyncratic}}.
-# The following stronger property implies ''m''-block-balancedness for ''m'' &ge; 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(<i>w'</i>) = len(''w'') + ''m'' + 1 satisfies |<i>w'</i>|<sub>''a''</sub> &ge; |''w''|<sub>''a''</sub> + 1.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word is a proof that for binary scales, balanced implies 1-chain-balanced.
+# The following stronger property implies ''m''-block-balancedness for ''m'' &ge; 1 but is not equivalent to it unless ''m'' = 1: ''s'' is ''m''-'''chain-balanced'''{{idiosyncratic}} if for every letter ''a'' in ''s'' and every factor of ''s'' of the form ''awa'', any factor ''w' '' in ''s'' such that len(<i>w'</i>) = len(''w'') + ''m'' + 1 satisfies |<i>w'</i>|<sub>''a''</sub> &ge; |''w''|<sub>''a''</sub> + 1.<ref>Sano, S., Miyoshi, N., & Kataoka, R. (2004). m-Balanced words: A generalization of balanced words. Theoretical computer science, 314(1-2), 97-120.</ref> (Compare [[recursive structure of MOS scales|MOS chunks]]; proving that the chunk sizes in a MOS themselves form a MOS word proves that for binary scales, balanced implies 1-chain-balanced.)
 == References ==
 [[Category:Scale]][[Category:Terms]]
 [[Category:Combinatorics on words]]