13 problems

Let $\kappa_{2,1}$ denote the Oldenburger–Kolakoski word over $\{1,2\}$, and interpret the frequency of a letter as its limiting proportion among prefixes, when that limit exists.…

Kolakoski sequence density conjecture. The limiting density of $1$ in the Kolakoski sequence equals

Kolakoski shuffle-square prefix problem. Does at least one prefix of the Kolakoski sequence form a shuffle square?

Let $u$ and $w$ be two subrows of the Kolakoski sequence, and let $P_n(u)$ denote the parity-history integral associated with $u$ at index $n$. Parity-history uniqueness conjecture…

Let $S$ be the Kolakoski sequence and let $\mathcal{SR}(S)$ denote its set of subrows. An $\infty$-regular subrow is a subrow that is $k$-regular for every $k$. Nonexistence conjec…

Generalized Uniformness Conjecture. The asymptotic density of $S$ equals $2^{-k-1}$.

Classical uniformness conjecture. The limiting density of $1$ in $K(1,2)$ equals $1/2$.

Let $n$ be an even integer, possibly nonpositive, and let $j>0$. Set … Let $t$ be either the sequence $(-1)v^{j-1}$ or the sequence $v^{j-1}n$; both have length $2j-1$. The map…

Maximal-orbit conjecture for $C_{1,n}$. The orbit of $t$ under the map $C_{1,n}(1,-)$ has length $2^j$. The general orbit bound proved in the source shows that this is the maximum…

Let $E_{m,n}$ and $C_{m,n}$ be the functions associated with iterated run-length expansion. For even $n>0$ and $j>0$, consider the finite sequence…

Dekking's conjecture. There is a suitable positive constant $c$ such that

Let $K$ be the Kolakoski sequence over the alphabet $\{1,2\}$. For each positive integer $n$, let $P_K(n)$ denote the number of subwords (factors) of length $n$ occurring in $K$, a…

Let $K$ be the Kolakoski sequence over the alphabet $Sigma=\{1,2\}$. A square is a word of the form $uu$ for a nonempty finite word $u$, and a sequence is cube-free if it contains…