Balanced word: Difference between revisions

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where |''u''|''x''''i'' is the number of occurrences of the letter ''x''''i'' in the word ''u''.

== Properties ==

* A balanced word or necklace in ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>.

* A balanced word or necklace on ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>.

* If ''w'' is an aperiodic infinite balanced word, then ''w'' is constructed via a finite sequence of ''congruence substitutions'' from infinite balanced words on 2 letters.<ref>Brauner, N., Crama, Y., Delaporte, E., Jost, V., & Libralesso, L. (2019). Do balanced words have a short period?. Theoretical Computer Science, 793, 169-180.</ref>

** A congruence word is a word u where the set of occurrences of each letter m in u is of the form <math>\{a_m n + b_m : n \in \mathbb{N}\},</math> for integers a_m and b_m.

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 where |''u''|<sub>''x''<sub>''i''</sub></sub> is the number of occurrences of the letter ''x''<sub>''i''</sub> in the word ''u''.
 == Properties ==
-* A balanced word or necklace in ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>.
+* A balanced word or necklace on ''N'' letters has a [[maximum variety]] bound of <math> N \choose {\lceil N/2 \rceil}</math>.
 * If ''w'' is an aperiodic infinite balanced word, then ''w'' is constructed via a finite sequence of ''congruence substitutions'' from infinite balanced words on 2 letters.<ref>Brauner, N., Crama, Y., Delaporte, E., Jost, V., & Libralesso, L. (2019). Do balanced words have a short period?. Theoretical Computer Science, 793, 169-180.</ref>
 ** A congruence word is a word u where the set of occurrences of each letter m in u is of the form <math>\{a_m n + b_m : n \in \mathbb{N}\},</math> for integers a_m and b_m.