Balanced word: Difference between revisions

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== Properties ==

* 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>

* 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.

** 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.

** A congruence substitution involves replacing the ''k''th occurrence of a fixed letter ''j'' in ''w'' with the ''k''th letter of ''u'' where ''u'' is a congruence word over a set of letters disjoint from that of ''w'' for all positive integers ''k''.

** A ''congruence substitution'' involves replacing the ''k''th occurrence of a fixed letter ''j'' in ''w'' with the ''k''th letter of ''u'' where ''u'' is a congruence word over a set of letters disjoint from that of ''w'' for all positive integers ''k''.

== Generalizations ==

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 == Properties ==
 * 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>
+* 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.
+** 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.
-** A congruence substitution involves replacing the ''k''th occurrence of a fixed letter ''j'' in ''w'' with the ''k''th letter of ''u'' where ''u'' is a congruence word over a set of letters disjoint from that of ''w'' for all positive integers ''k''.
+** A ''congruence substitution'' involves replacing the ''k''th occurrence of a fixed letter ''j'' in ''w'' with the ''k''th letter of ''u'' where ''u'' is a congruence word over a set of letters disjoint from that of ''w'' for all positive integers ''k''.
 == Generalizations ==