21 problems

Let $D(r,c,d)$ denote the number of ways to place $d$ non-bonding dominoes on an $r\times c$ rectangular board. Quadratic formula conjecture. For integers $r,c\ge 3$, … The formula…

Let $p_a$, $p_b$, $p_c$, and $p_d$ be the edge-density quantities associated with the dimer model on an $n\times n$ torus in the thermodynamic limit. Consider an infinite sequence…

Generalized product formula conjecture. There exists a sequence of polynomials $o_n(k,t)\in\mathbb{Z}[k,t]$ such that

Consider a sequence of finite regions converging to a fixed shape, and let $(s,t)$ be the tilt given by the variational principle wherever that tilt is defined. Let $\mu_{s,t}$ be…

Let $\mu_{s,t}$ denote the measure on domino tilings of the plane associated with tilt $(s,t)$. A measure is conditionally uniform when, given a tiling outside any finite region, t…

Let $(s,t)$ satisfy $|s|+|t| \leq 2$, and let $a,b,c,d$ be weights satisfying $ab=cd$ such that the average height function for weighted torus tilings has tilt $(s,t)$. Let…

In the dimer model on an $n \times n$ torus, let $p_a$, $p_b$, $p_c$, and $p_d$ denote the limiting densities of the four edge types in the thermodynamic limit as…

Modulo-$4$ conjecture. Except sometimes when $r$ and $c$ are both congruent to $3$ modulo $4$, the count $a(r,c;h)$ is a multiple of $4$.

Kong's modulo-$8$ conjecture. For all $k$, the integer $c_k$ is congruent to $1$ modulo $8$.

Let $\bar d(r,c)$ denote the maximum number of non-bonding dominoes that can be placed on an $r\times c$ rectangular board. Maximum-filling conjecture. If one of $r$ and $c$ is an…

Let $AD(n)$ be the Aztec diamond of order $n$. Let $O(2n,[2n])$ be the set of off-diagonally symmetric domino tilings of $AD(2n)$ with no boundary defect, and let $D(2n-1)$ be the…

Di Francesco's conjecture.

A simply connected region is a domino-tileable region in the setting of the paper whose interior has no holes. Let the domino-tiling counting theorem of the paper be denoted by The…

For each $n\geq 1$, let $Z^{20V}_n$ denote the total number of 20-vertex configurations with DWBC3 boundary conditions on the quadrangle ${\mathcal Q}_n$, and let $Z^{DT}_n$ denote…

Consider the cubiculated box … Let $\mathbf T,\mathbf T_0,\mathbf T_1$ be independent uniformly distributed random tilings of $\mathcal R$, and let $\operatorname{Tw}$ denote the t…

For nonnegative integers $a,b,c$, let $N_{a,b,c}$ denote the number of 20V configurations on the rectangular grid of size $(a+b+1)\times(b+c+1)$ with DWBC4-type boundary conditions…

Let $P_{n,k}$ be the pentagonal domain carrying the extended DWBC3 boundary conditions, and let ${\mathcal T}_{n,k}$ be the corresponding extended triangle whose domino tilings are…

Let $B_n$ be the sequence associated with 20V configurations with DWBC3 boundary conditions, and let $b_n=T({\mathcal T}_n)$ denote the number of domino tilings of the triangle…

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a triangle removed from the top corner at height $N/\sqrt{2}+2^{-5/6}N^{1/3}R$. Define … For given observation…

Let $X_N^R$ denote the top line of an Aztec diamond of size $N$ with a square removed from the top so that its lower tip is at height $N/\sqrt{2}+2^{-5/6}N^{1/3}R$. Define … For an…

A semicontiguous minor is a minor $M_A^B$ in which only one of $A$ or $B$ is contiguous. Domino-tiling conjecture. Every semicontiguous minor has an associated domino-tiling region…