57 problems

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 $r(\mathcal{P})$.

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…

Ollinger's conjecture. The $8$-polyomino tiling problem is undecidable.

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…

Let $d$ be a positive integer and let $n$ be the number of unit $d$-dimensional hypercubes in a facet-to-facet connected system, called a $d$-dimensional polyomino. Its convex hull…

Regularity–rook-number question. Is the regularity of $\mathbb{K}[\mathcal{P}]$ always equal to $r(\mathcal{P})$ for any polyomino $\mathcal{P}$?

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

For $k$-convex polyominoes, let $Q_k(x)$ denote the generating function by semi-perimeter. Let $U_k(t)$ and $T_k(t)$ be the Chebyshev polynomials of the second and first kinds, res…

A $k$-convex polyomino is a convex polyomino in which any two cells can be joined by a directed internal NE, NW, SE, or SW path with at most $k$ right-angle bends. Let $Q_k(x)$ be…

Conjecture. The growth constant satisfies

Let $G$ be a polyomino, and let $H$ and $K$ be two distinct Hamiltonian cycles of $G$. A valid quadruple-switch move consists of four successive switches whose final subgraph is ag…

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…

A polyomino is a connected union of unit squares in the square lattice, with cuts between squares permitted; it is cube-foldable if it can be folded along its edges to cover the su…

Let $P$ be a connected bounded region in the plane formed out of $2 \times 2$ blocks joined by their sides. A minimum-turn Hamiltonian cycle of $P$ is a Hamiltonian cycle using the…

Beauquier–Nivat conjecture. The $2$-polyomino tiling problem is decidable.

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…

Let a labelled parallelogram polyomino of size $m\times n$ be a pair of northeast lattice paths on an $m\times n$ grid, with labels on the cells adjacent to the prescribed boundary…

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 $\mathcal{P}$ 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…

Let $g_L$ be the gerrymander sequence for $2L\times2L$ boards, and let $\widehat{g}_L$ be the generalised gerrymander sequence for $L\times L$ squares. Asymptotic comparison conjec…

Let $G_L(1)/2$ count generalised gerrymander configurations on an $L\times L$ square, and let $\lambda$ be their proved leading growth constant. Write the subdominant correction us…