188 problems

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…

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…

For a finite alphabet $\mathcal A$, determine the spectrum $\operatorname{Spec}_{\mathrm{rD}}(\mathcal A)=\{\operatorname{rD}(w):w\in\mathcal A^{\mathbb N}\}$ of the refined Diopha…

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above. The critical exponent conjecture for…

Let $w$ be an infinite word. A prefix of arbitrarily high palindromic length means that for every integer $M$ there is a prefix $u$ of $w$ with $\operatorname{PL}(u)\geq M$. Frid–P…

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above, and let $\rho_{{\bf x}_k}(n)$ denote its s…

Mäkelä's conjecture. There exists an infinite ternary word whose only abelian square factors are $00$, $11$, and $22$.

An extremal square-free word is a square-free word such that inserting any letter in any position introduces a square. The nonexistence conjecture. There are no extremal square-fre…

Nonchalant-word infinitude conjecture. The sequence of nonchalant words over $\mathbb A_n$ is infinite for every $n\geqslant 3$.

Let $\mathbf u$ be a binary generalized pseudostandard word, and let $\mathcal C_{\mathbf u}(n)$ denote its complexity, namely the number of factors of length $n$ in $\mathbf u$. T…

Let $a_k(n)$ denote the number of distinct subword complexity sequences of length $n \ge 1$ over a $k$-letter alphabet. Recurrence conjecture. There exists a function $f(k)$ such t…

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…

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…