8 problems

Large-alphabet shuffle-square growth conjecture. As a function of $k\in\mathbb{N}$,

Binary shuffle-square density conjecture. As $n\to\infty$, asymptotically half of all binary words of length $2n$ are shuffle squares; equivalently,

Cutting-distance conjecture. For each $k\geqslant2$, every even $k$-ary word $W$ satisfies

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

Bukh–Borisov deletion-distance conjecture. There is a constant $c$ such that, for every $n$,

For a binary even word $W$ of length $2n$, let $s(W)$ be the number of circular shifts of $W$ that are shuffle squares. Let $s_{2n}$ be the minimum of $s(W)$ over all binary even w…

Fix an alphabet of size $k$. Let $E_n$ be the set of all even $k$-ary words of length $n$, and let $S_n$ be the set of all permutations of $[n]$. Define $m_k(n)$ to be the minimum…

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…