6 problems

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 $2^{-k-1}$.

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 $K=(K_n)_{n=1}^{\infty}$ be the classical Kolakoski sequence on the alphabet $\{\mathtt{1},\mathtt{2}\}$, and let $o_n=|\{i:K_i=\mathtt{1},\ 1\leq i\leq n\}|$ be the number of…