17 problems

Strong Wilf-equivalence conjecture. Two binary words $p$ and $q$ are strongly Wilf-equivalent if and only if they are trivially equivalent.

Let a binary word be an infinite word over the alphabet $\{0,1\}$. A square is a nonempty word of the form $xx$, and a word is cubefree if it contains no nonempty factor of the for…

Uniqueness conjecture for 2-full words. The 2-full words $W\in\mathcal{MW}^{\leq 2}_{h\times w}$ are unique up to symmetries of the plane and a local symmetry (a flip) on hook snak…

Two-cut conjecture. Every even binary word $W$ satisfies

Characterization problem for doubly binary even words. Characterize all doubly binary even words that are not shuffle squares.

He–Huang–Nam–Thaper conjecture. Almost all even binary words are shuffle squares.

Monotonicity conjecture. The number of pairs $(v,w)$ satisfying

Logarithmic-length conjecture. For every integer $n\geqslant 1$, there exists a binary word $\mathfrak{w}$ such that

Let $(a,b)$ be a pair of positive integers with $1\leq a<b$. An infinite binary word has pseudoperiod $(a,b)$ if each position is compatible with a repetition having periods $a$ an…

Assuming the growth-rate limit exists, define … for $r \in \mathbb{R}^+$. Monotonicity and discontinuity conjecture. The function $r \mapsto L(r)$ is increasing on $[0,+\infty)$ an…

For each positive real number $r$, let $\mathcal{W}_{r,n}$ be the set of restricted binary words of length $n$ with parameter $r$. Existence conjecture. For given…

Let $\mathcal{W}_{q,n}$ denote the set of restricted binary words of length $n$ associated with the positive rational parameter $q$. A 1-Gray code is an arrangement of all words in…

Let $n\ge 1$, and let $s$ be chosen uniformly from the binary words $\{0,1\}^{2n}$ having an even number of ones. A typical-word shuffle-square conjecture. With high probability as…

Maximum-cardinality conjecture. This is the maximum cardinality: equivalently, the bound $2^k-2$ in the stated theorem can be replaced by

Let $n$ be a nonnegative integer, and let binary words of length $2n$ mean words formed from the alphabet $\{0,1\}$ with $2n$ positions. Hivert–Reiner–Rosas conjecture. The number…

Let $1\leq p<q$ be coprime integers. For a word $w\in\mathbb{W}_{p,q}$, let $\mathcal{I}(w)=(w_1,\ldots,w_q)$ be its base-2 orbit and define the partial products by … The relation…

Let $x$ and $y$ be distinct binary words of length $2n$. For a word $w$, let $F_n(w)$ denote its set of length-$n$ factors, and let $\pi(w)$ denote the parameter defined in the pap…