Balanced word

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 x_i 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].
• If w is an aperiodic infinite balanced word, then w is constructed via a finite sequence of "congruence substitutions" beginning with a Sturmian word.
• Some periodic balanced words are not of the above form. For alphabets of size d = 3, ..., 7, the only exception up to substituting letters via permutation is the Fraenkel word F_d, defined via [math]\displaystyle{ F_0 = \mathbf{0}, F_n = F_{n-1} \mathbf{n} F_{n-1}. }[/math]