17 problems

Let ${\bf x}=(a_i)_{i\geq 0}$ be an infinite word over a finite alphabet $\Sigma$. Write $\rho_{\bf x}(n)$ for the number of distinct length-$n$ factors of ${\bf x}$, and let…

Sing's power-law conjecture. Over $\mathpzc{A}=\{a,b\}$,

Let $\mathpzc{A}=\{a,b\}$ be a binary alphabet, let $\mathcal{C}_f^\infty$ be the set of finite f-smooth words over $\mathpzc{A}$, and let $p_{\mathcal{C}_f^\infty}(n)$ denote its…

Let $\mathcal{C}_f^\infty$ be the set of finite f-smooth words over the binary alphabet $\{1,2\}$, and let $p_{\mathcal{C}_f^\infty}(n)$ denote the number of factors of length $n$…

Let ${\mathbf G}=\gamma^\omega(0)$, where $\gamma=[01,2,02]$, let $h$ be a morphism in Table, and let $h^R$ denote the reversed morphism used in that table. Let $\mu$ be the critic…

Let ${\mathbf G}=\gamma^\omega(0)$, where $\gamma=[01,2,02]$, and let $\mu$ be its critical exponent. For each morphism $h$ in Table, consider the infinite word $h({\mathbf G})$. M…

Let $\gamma=[01,2,02]$ and let ${\mathbf G}=\gamma^\omega(0)$ be its fixed point. Its factor complexity is $2n+1$, and its critical exponent is $\mu=2+\frac{1}{\lambda^2-1}$, where…

Let ${\bf G}$ be the infinite ternary word described in the source, with critical exponent $2.4808627\cdots$ and factor complexity $2n+1$ for every $n\geq 1$. Shur's conjecture. Th…

For a positive integer $n$, let $fann$?

Let $k\geq 1$, and let $\mathbf{x}_k$ be the Thue-Morse-like sequence obtained from the family described in the source. For a binary sequence, its factor complexity $p(n)$ is the n…

Let $A$ be an alphabet of size $N$, let $\sigma$ be a cyclic permutation of $A$, and let $W_{A, \sigma, a_0}$ be the infinite word generated by the cyclic shift morphism…

Complexity bound conjecture. The complexity function $p_\mathcal{L}$ for any $S(e,23,e)$-adic language $\mathcal{L}$ satisfies

Let $\mathbf{x}$ be a sequence, and let $r_{\mathbf{x}}(n)$ and $\rho_{\mathbf{x}}(n)$ denote its reflection complexity and factor complexity, respectively. The half-or-full limit…

Let $\mathbf{x}$ be a sequence of at most linear factor complexity, meaning that its factor-complexity function grows at most linearly. The bounded-difference conjecture. … is boun…

Let $\mathbf{x}$ be a sequence. The reflection complexity periodicity conjecture. … if and only if $\mathbf{x}$ is eventually periodic. One direction is known: if $\mathbf{x}$ is e…

Let $f53f$ be a morphic sequence, and let $p_{\mathbf{w}}(n)$ denote its factor-complexity function. Morphic-sequence complexity spectrum conjecture. The function $p_{\mathbf{w}}$…

Let $f53f$ be a sequence over a finite alphabet. An $S$-adic sequence is one admitting an $S$-adic representation for some set $S$ of morphisms; in particular, the representation i…