15 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

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

Kolakoski shuffle-square prefix problem. Does at least one prefix of the Kolakoski sequence form a shuffle square?

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

Four-avoidability conjecture for shuffle squares. Shuffle squares are $4$-avoidable.

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

For each length, consider the set of even binary words and the subset consisting of shuffle squares. The asymptotic binary shuffle-square conjecture. As the length tends to infinit…

A binary word uses an alphabet of two letters, and an even word is a word in which every letter occurs an even number of times. A word is a cyclic shuffle square if it splits into…

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…

An even word is a word in which every letter occurs an even number of times. A ternary word uses an alphabet of three letters. A dihedral shuffle square is a word that can be split…

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…