6 problems

Let the sum range over all unlabeled complete binary trees $T$ with $n$ internal vertices, let the product range over all internal vertices $v$ of $T$, and let $h_v$ be the number…

Let $m,\ell \in \mathbb{Z}_{\geq 1}$ and let $\lambda,\mu \in \mathcal{P}_{m,\ell}$ satisfy $\lambda \supset \mu$. Put … Here $f^{\mathring{\lambda}/\mathring{\mu}}$ denotes the nu…

Let $\lambda$ be a partition of $n$, and let $\mathcal{H}(\lambda)$ be its multiset of hook lengths. Let $\mathcal{H}(\lambda)^{\diamond}$ be the multiset of hook lengths with triv…

Shifted skew-shape product conjecture. One has

The d-complete poset hook-length bijection conjecture. If $G\in\mathcal{G}_c^0$, then $s(G)=c$ and $e(G)=G$; if $G\in\mathcal{G}_c^k$ for $k\geq1$, then…

Let $P$ be a connected $d$-complete poset with maximum element $v_0$, rank function $r:P\to\mathbb{N}$, top tree $T$, and $d$-complete coloring $c:P\to T$. For a $P$-partition…