6 problems

Rauzy's conjecture. Except for very particular vectors of letter frequencies, there do not exist any infinite ternary words with constant abelian complexity equal to $3$.

Let ${\bf{t}}$ be the Tribonacci sequence, fixed point of $0\mapsto 01$, $1\mapsto 02$, $2\mapsto 0$. For each $k\ge 1$ and $n\ge 0$, let $\rho^{k}_{\bf{t}}(n)$ denote its generali…

Automatic-sequence regularity conjecture. The $k$-abelian complexity of any $\ell$-automatic sequence is an $\ell$-regular sequence.

Let $k\ge 2$ and let $\ell\ge 1$ be integers. A sequence is $k$-automatic if it is generated by a finite automaton reading the base-$k$ representation of its indices. For an infini…

Let $k\geq 2$ be an integer, and let $w$ be a $k$-automatic infinite word. Its $2$-abelian complexity is the sequence counting the equivalence classes of factors of each length und…

Let $w$ be an infinite recurrent word over a $4$-letter alphabet, where recurrent means that every factor of $w$ occurs infinitely often in $w$. For each positive integer $m$, let…