14 problems

Let $g(n,k,t)$ be the number of $n$-letter words over the alphabet $[k]$ having exactly $t$ weakly outstanding elements, and define … Let $\Delta_x$ be the forward difference opera…

Let $[n]^n(\pi)$ denote the set of words of length $n$ over the alphabet $[n]=\{1,2,\ldots,n\}$ that avoid a permutation pattern $\pi$, and let $S_n(\pi)$ denote the set of permuta…

Shortest common superpattern conjecture. For any $l\ge 1$,

Layered extremal word conjecture. If $\Pi$ is a set of layered patterns, then

Folded-form tightness conjecture. The construction's upper bound is tight:

Optimality conjecture. The constructed sequence is asymptotically optimal: the limit

For $k\geq 1$ and an alphabet size $r$, let $f(k;r)$ be the minimum $n$ such that there is a word $\sigma\in [r]^n$ containing every permutation in $S_k$ as a pattern; such a word…

Conjecture on extremal abelian square-free words. There are infinitely many extremal abelian square-free words over a four-letter alphabet.

Let $P$ be an avoidable pattern, meaning that some alphabet admits arbitrarily long words avoiding $P$. For a fixed alphabet, an extremal $P$-avoiding word is a word that avoids…

For each positive integer $m$, let $\mathcal E_m$ be the set of exceptional normalized words of length $m$, and let $\mathcal{NW}_m$ be the set of normalized words of length $m$. H…

Let $w$ be a word, and let $\langle w\rangle_{\operatorname{\mathsf{hare}}}$ and $\langle w\rangle_{\operatorname{\mathsf{tortoise}}}$ denote its hare- and tortoise-fertilities. Th…

Let $w$ be a word of length $m$, and let $\langle w\rangle_{\operatorname{\mathsf{hare}}}$ and $\langle w\rangle_{\operatorname{\mathsf{tortoise}}}$ denote its hare- and tortoise-f…

Henshall–Rampersad–Shallit conjecture. The number of square binary words of length $2n$ is

Let $\beta$ be a non-empty permutation. For permutations $231\oplus\beta$ and $312\oplus\beta$, Wilf-equivalence for words means that the two patterns are avoided by the same numbe…