6 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…

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

Joint equidistribution conjecture. The pairs

Bijection conjecture. There is a bijection between beta$(0,1)$-trees on $n$ nodes fixed under $g$ and beta$(0,1)$-trees on $\lfloor n/2\rfloor+1$ nodes with one distinguished exces…

Let $\lambda_1$ and $\lambda_2$ be Dyck paths whose corresponding trees are isomorphic when their embeddings in the plane are ignored. For a Dyck tiling…