4 problems

Let $R(n,m)$ be the set of positive integers $k$ for which some word of length $2n$ over a complementary alphabet of size $m$ has exactly $k$ valid rooted plane trees. Black–Drelli…

Let $A$ be a complementary alphabet of size $m$, and let $R(n,m)$ be the set of positive integers $k$ for which there is a word $P$ of length $2n$ over $A$ having exactly $k$ $P$-v…

Universality conjecture. For all $k \in \mathbb{Z}_{\geq 0}$ there exists $n,m$ such that

Alphabet-independence conjecture. The set $R(n,m)$ depends only on $n$.