8 problems

Let $\mathfrak{F}^{(a,b)}_n$ denote the standard length-$n$ prefix or level-$n$ finite word in the biperiodic Fibonacci construction, and let $\Phi$ be the palindromisation operato…

Let $\mathfrak{F}^{(a,b)}$ be the infinite binary biperiodic Fibonacci word with parameters $a,b\geqslant 1$. There is an explicit, effectively computable map $\theta$ on infinite…

Let a Fibonacci word be a finite word in the Fibonacci-word sequence, and let a net occurrence be an occurrence of a repeated string with unique left and right extensions. Fibonacc…

Multiplicity-matrix conjecture. The matrix $\mathrm{\bf N}_n$ is upper triangular. Furthermore, if $N_{u,v}$ is nonzero, then $u\succeq v$ and $\mathcal{h}(u)=\mathcal{h}(v)$. This…

Let $n\geq 0$ and $q\geq 1$, and let $\mathcal{W}^q_n$ denote the set of Fibonacci $q$-decreasing words of length $n$. A 1-Gray code for $\mathcal{W}^q_n$ is an ordering of its wor…

Let $\mathbf{f}^{(2)}$ denote the Fibonacci word, and let $\overline{\mathrm{A_N}}(\mathbf{f}^{(2)})$ be its limiting nondeterministic automatic complexity rate. Let…

Minimal-$xxx^R$ enumeration conjecture. For $n=F_{3k+1}$ there are $4$ such words of length $n$. For $n=F_{3k+1}\pm F_{3k-2}$ there are $2$ such words. Otherwise there are none. Fo…

Primitive-period classification conjecture. If $z^\omega$ avoids $xxx^R$, then