9 problems

Let $p_d(n)$ denote the number of $d$-dimensional partitions of $n$. For solid partitions, $d=4$, and the relevant normalization is $n^{1-1/d}$. MacMahon's numbers are the numbers…

Curve-vertex sign rule. The sign relative to the square root $\mathsf{v}_pi^{i\prime}$ is $(-1)^{\sigma_i(\pi)}$. This is a proposed orientation rule for curve-like toric Calabi–Ya…

Point-vertex sign rule. The sign relative to the square root $\mathsf{v}_pi^i$ is $(-1)^{\sigma_i(\pi)}$. This proposes an explicit orientation rule for the toric Calabi–Yau $4$-fo…

Equality conjecture for solid-partition weights. One has

Polynomial specialization conjecture. The following properties hold: (a) $L_\pi(0,0,0,-d)\mathsf{w}_\pi$ has no pole at $\lambda_4=-(\lambda_1+\lambda_2+\lambda_3)$; (b) its specia…

Uniqueness conjecture for affine DT4 weights. There exists a unique way of choosing the signs of the equivariant weights $\mathsf{w}_\pi$ such that the affine DT4 vertex identity h…

The affine DT4 vertex conjecture. There exists a way of choosing the signs of the weights $\mathsf{w}_\pi$ such that

A solid partition is a three-dimensional partition, and let $z$ mark its size. MacMahon's conjecture. The generating function for solid partitions is … This conjecture was disprove…

For given values of $B$ and $n$, let $p_3(B,n)$ be the number of solid partitions of $n$ such that all coordinates of all nodes are at most $B$. Define the partial entropy by … Con…