11 problems

Structure conjecture. For all $i\in\{1,2,\ldots,|I|\}$ and $k\in\mathbb{Z}_{>0}$,

Let $\widehat{\Delta}^{+,\mathrm{re}}$ be the set of positive real roots of affine type $A^{(1)}_n$, and let $<$ be the order defined by. Pre-convexity conjecture. The order satisf…

A Lyndon word over $\{0,1\}$ is a word that is the unique minimum among all of its rotations. For $k\geq2$, let $\alpha=(1,0,0,\dots,0)$ be the prefix of length $k$, and let…

A Lyndon word over $\{0,1\}$ is a word that is the unique minimum among all of its rotations. Let $\alpha$ be a fixed prefix, and let…

Coin Arrangements identity. For any $N\geq n>1$,

Let $\Delta^+$ be the set of positive roots, let $d\in\mathbb{Z}$, and let $\ell(\alpha,d)$ denote the Lyndon loop word of degree $(\alpha,d)$. Let $\Phi^L$ be the homomorphism fro…

Let $A$ be a finite alphabet, and let $w$ be a finite generalized Lyndon word over $A$. The length of $w$ is denoted by $|w|$. Extendability conjecture. Every finite generalized Ly…

Let $A$ be an alphabet, and let $A^\omega$ denote the set of infinite words over $A$. A generalized Lyndon word is a finite or infinite word that is Lyndon with respect to a genera…

Saari's conjecture. If $n\ne 6$ and $w$ is minimal, then $w$ is a Christoffel word.

Let $\ell(n)$ be the minimum number of Lyndon factors among Lyndon words of length $n$: … A minimizer conjecture. If $w$ is a Lyndon word with $|w|\neq 6$ and…

Let $\zeta(s_1,\ldots,s_k)$ denote a multiple zeta value of weight $s_1+\cdots+s_k$. A word in the alphabet $\{2,3\}$ is called Lyndon when it is strictly smaller than each of its…