Balanced word
Let d ≥ 0. A scale word s is d-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 d, }[/math] where |u|xi is the number of occurrences of the letter xi in the word u. A scale is balanced if it is 1-balanced.