4 problems

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…

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…

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…