11 problems

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

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

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

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…

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…