6 problems

Let $\mathcal{P}$ be a collection of cells. The Gorenstein characterization conjecture asserts that the following are equivalent: 1. $K[\mathcal{P}]$ is Gorenstein; 2. the $h$-poly…

Domino-stability conjecture. The ring $K[\mathcal{P}]$ is Gorenstein if and only if $\mathcal{P}$ is domino-stable.

Rinaldo–Romeo conjecture. The polyomino $\mathcal{P}$ is thin if and only if

Let $cmathcal{P}$ be a simple polyomino, meaning a polyomino without holes, and let $K[cmathcal{P}]$ be its associated coordinate ring. Qureshi–Rinaldo–Romeo conjecture. The switch…

Let $\mathcal{P}$ be a polyomino, let $K$ be a field, and let $K[\mathcal{P}]$ denote its coordinate ring. A zig-zag walk is the type of walk in a polyomino referred to in the sour…

Let $w$ be a permutation, and let the counting function for matrices of rank $r$ whose support avoids the permutation diagram $O_w$ be the associated $q$-analogue of placements of…