Balanced word: Difference between revisions
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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''. | 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 == | == Properties == | ||
* A balanced word or necklace on ''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> | * 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 == | == Generalizations == | ||
Revision as of 12:43, 31 December 2023
Let d ≥ 0. A word or necklace s is balanced if its balance satisfies the following:
[math]\displaystyle{ \operatorname{balance}(s) := \max \big\{ \big| |w|_{x_i} - |w'|_{x_i} \big| : x_i \text{ is a letter of }s\text{ and }k = \operatorname{len}(w) = \operatorname{len}(w') \big\} \leq 1, }[/math]
where |u|xi is the number of occurrences of the letter xi in the word u.
Properties
- A balanced word or necklace on N letters has a maximum variety bound of [math]\displaystyle{ N \choose {\lceil N/2 \rceil} }[/math].