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

TL;DR

This paper investigates the extremal properties of the words $0^{q-p}1^p$, showing they are the most unbalanced words with maximal partial sum orbit and minimal product, contrasting with balanced words.

Contribution

It establishes the extremal orbit properties of the words $0^{q-p}1^p$, demonstrating their role as the most unbalanced words in terms of partial sum and product.

Findings

Orbit of $0^{q-p}1^p$ is the greatest in partial sum order.

These words have the smallest product among their orbit set.

They are the most unbalanced words with extremal properties.

Abstract

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.