The most unbalanced words $0^{q-p}1^p$ and majorization

A finite word $w\in\{0,1\}^*$ is balanced if for every equal-length factors $u$ and $v$ of every cyclic shift of $w$ we have $||u|_1-|v|_1| <= 1$. This new class of finite words were defined in [JZ]. In [J], there was proved several results considering finite balanced words and majorization. One of the main results was that the base-2 orbit of the balanced word is the least element in the set of orbits with respect to partial sum. It was also proved that the product of the elements in the base-2 orbit of a word is maximized precisely when the word is balanced. It turns out that the words $0^{q-p}1^p$ have similar extremal properties, opposite to the balanced words, which makes it meaningful to call these words the most unbalanced words. This article contains the counterparts of the results mentioned above. We will prove that the orbit of the word $u=0^{q-p}1^p$ is the greatest element in the set of orbits with respect to partial sum and that it has the smallest product. We will also prove that $u$ is the greatest element in the set of orbits with respect to partial product.

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