28 problems

Let $R\in\mathbb{N}^p$, $C\in\mathbb{N}^q$, and let $\mathbf{M}\in\mathcal{M}_{p,q}(\{-1,0,1\})$. For each integer $i$, let $\mathcal{P}_i(\mathbf{M})$ denote the set of cells at l…

Let $n$ congruent regular unit hexagons form an edge-to-edge connected system. Hexagon polyomino area bound. The area of its convex hull is at most … The statement is presented in…

Let $P$ be a $d$-dimensional polyomino with parameters $l_1,\dots,l_d$ and $v_1,\dots,v_d$, and suppose that the convex hull of $P$ has maximum volume. Let $P'$ be a subpolyomino o…

Switching-rook-polynomial conjecture. The $h$-polynomial of $\mathbb{K}[\mathcal{P}]$ is equal to $\widetilde{r}_{\mathcal{P}}(t)$, and its regularity is equal to $r(\mathcal{P})$.

Mascia et al.'s conjecture. $\mathcal{P}$ is prime if and only if $\mathcal{P}$ contains no zig-zag walk.

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

Let a $(begin{tikzpicture}[scale=.25] draw (1,0)--++(.5,0)--++(0,.5)--++(-1.5,0)--++(0,-.5)--++(1,0)--++(0,1)--++(-.5,0)--++(0,-1); end{tikzpicture},n)$-prismatic polyomino be a pr…

For a fixed positive integer $k$, the $k$-polyomino tiling problem asks whether there is an algorithm deciding whether a set of $k$ polyominoes can tile the plane by translated cop…

Let $\mathcal{P}$ be a simple thin polyomino, meaning a path polyomino with no interval containing more than two cells, and suppose that every maximal interval consists of a single…

Let $P(n)$ denote the number of polyominoes with $n$ cells. Successive-ratio monotonicity conjecture. The sequence … is increasing. The conjecture is motivated by the available val…

Let $P(n)$ be the number of polyominoes with $n$ cells. A polyomino is constructible if it is the concatenation of two polyominoes of smaller sizes, and inconstructible otherwise;…

Let $P(n)$ denote the number of polyominoes with $n$ cells, and let … be Klarner's constant. Whittington–Soteros conjecture. There exist constants $A,T$ such that … This is a widel…

Convex-polyomino maximal queen domination conjecture. The maximal domination problem for non-attacking queens on convex polyominoes is NP-complete.

Convex-polyomino queen domination conjecture. The minimal domination problem for attacking, or non-attacking, queens on convex polyominoes is NP-complete.

Convex-polyomino rook domination conjecture. The minimal domination problem for attacking or non-attacking rooks on convex polyominoes is NP-complete.

Maximal independent queen domination conjecture. The maximal independent queen domination problem on polyominoes is NP-complete.

König-type conjecture. The ideal $I_{\mathcal{P}}$ is of König type.

Height–rank conjecture. One should have

Let $\mathcal{P}$ be a weakly connected collection of cells, and let $I_{\mathcal{P}}$ denote its polyomino ideal. A zig-zag walk is the particular sequence of inner intervals of…

Let $m,n\in\mathbb{N}$, let $\mathsf{LPP}(m+1,n+1)$ be the set of labelled parallelogram polyominoes of the indicated dimensions, and let $\mathsf{area}(P)$, $\mathsf{pmaj}(P)$ and…

Thinness conjecture. ${\mathbb P}$ is thin if and only if

Let $C_4$ denote a square cycle, and let $G^4_S$ be a square polyomino formed from $S$ copies of $C_4$. Write $V(G^4_S)$ and $E(G^4_S)$ for its vertex and edge sets, and let…

Primality conjecture. The following conditions are equivalent:

The pmaj formula. For $m\geq 1$ and $n\geq 1$,