187 problems

Factor-occurrence rationality conjecture. For every fixed $k \ge 3$ and every non-empty factor $B \sqsubseteq \mathbf{W}^{(k)}$, there exist an integer $s(B) \ge 1$ and a polynomia…

Let ${\bf f}=f_0f_1f_2\cdots$ be a paperfolding word over $\{1,4\}$, and define ${\bf v}=v_0v_1v_2\cdots$ by … for all $n\geq 0$, with the even-position symbols of ${\bf f}$ recode…

Let an infinite word be a sequence over a 4-letter alphabet. An $r$-power is a word consisting of repetitions of a nonempty block with exponent $r$, and a word avoids $r$-powers in…

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…

Let $T=(t_n)_{n\ge0}$ be the steering word considered in the paper, and let $p_T(k)$ denote the number of distinct factors of length $k$ occurring in $T$. Exact complexity conjectu…

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…

Let $w$ be a binary word. A balanced border is a border of $w$ whose numbers of $0$s and $1$s are equal; the statistics $\rho^w$ and $q^w$ are the letter-frequency quantities assoc…

Let $\Sigma$ be a finite alphabet, and let a morphism on $\Sigma$ be called abelian power-free when its images avoid abelian powers of the relevant exponent. Carpi's set of condition…

Let $f$ be the infinite Fibonacci word, let $C_f$ be its closed-rich constant, and let $\phi=\frac{1+\sqrt{5}}{2}$ be the golden ratio. Fibonacci closed-rich constant conjecture. F…

Let $f$ be the infinite Fibonacci word, let $\Sigma$ be its alphabet, and define … Here $\operatorname{Cl}(w)$ is the number of distinct closed factors of $w$, and $F_n$ denotes th…

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 $3$.

Effective-partition conjecture. The word $w$ contains at least

Fazekas–Mammoliti–Mercaş–Simpson conjecture. A binary word of length $n$ contains at least

Let $w,w'$ be two words, and let $\tilde{c}(w)$ denote the second auto-correlation constant defined by the asymptotic second moment of the number of leaves associated with $w$. The…

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

Let $\mathpzc{A}=\{a,b\}$ be a mixed binary alphabet, let $\mathcal{C}^\infty$ be the set of infinite smooth words over $\mathpzc{A}$, let $\mathcal{C}_f^\infty$ be the set of fini…

For each integer $n\ge 5$, let ${\bf A}_n$ be the $n$-letter alphabet, ${\bf A}_n^{\omega}$ the set of infinite words over it, ${\bf T}_n$ the class of boundary words, and let…

Square-free word conjecture. Given a sequence of alphabets $(\mathcal{A}_i)_{i\ge1}$ with $|\mathcal{A}_i|\ge3$ for all $i$, there exists an infinite square-free word that respects…

Thue list-number conjecture. The Thue list number of the infinite path is $3$.

Let $w$ be an Arnoux–Rauzy infinite word. Its $2$-binomial complexity and subword complexity are functions assigning to each positive integer the number of equivalence classes of f…

Let $\mathcal{C}_d$ be the class of $d$-ary sequences rich in palindromes. For each $n\in\mathbb{N}$, let $\mathbf{u}_n$ denote the sequence defined in the paper by the correspondi…

Let $W$ be a set of words, let $F(W)$ denote the associated object, and let $(<_A,<_D)$ be the pair of orders used in the order condition. Order-condition characterization conjectu…

Let $L$) be a language on $k$ letters satisfying a symmetric order condition and having no connection. Its palindromic complexity is the number of palindromic factors of each lengt…

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…