10 problems

Let $m$ be a nonnegative integer and let $N$ and $r$ be positive integers with $N\ge r+2$. Consider rhombus tilings of the two hexagons with side lengths $(N,2m,N,N,2m,N)$ and…

Let $m$ be a nonnegative integer and let $N$ and $r$ be positive integers with $N\ge r+1$. Consider rhombus tilings of the two hexagons with side lengths $(N,2m,N,N,2m,N)$ and…

Let $m$ be a nonnegative integer and let $N$ and $r$ be positive integers with $N\ge r$. Consider rhombus tilings of the two hexagons with side lengths $(N,2m,N,N,2m,N)$ and…

Consider a percolation process on a sufficiently regular rhombus tiling, with critical threshold $p_c$ and stable directions associated with the process. Stable-direction critical-…

Let $G=(V,E)$ be the adjacency graph of a uniformly repetitive (or uniformly recurrent) rhombus tiling. Let $F$ be a percolation process, monotone and freezing, on configurations…

Let $D_{s,t}(n)$ denote the determinant associated with the corresponding family of cyclically symmetric rhombus-tiling regions, with the parity of $s-t$ distinguishing ordinary ti…

Let $\mu$ be an indeterminate, let $n\geqslant1$ and $r\geqslant0$ be integers, and define the rising factorial $(a)_k=a(a+1)\cdots(a+k-1)$. Let $D_{-1,2r}(2n)$ be the determinant…

Let $D_{1,1}(n)$ be the determinant appearing below, with $\binom{\bullet}{\bullet}$ denoting a binomial coefficient, $\binom{\bullet}{\bullet}$ and $\binom{\bullet}{\bullet}$ inte…

MacMahon's enumeration conjecture. The number of vertically symmetric rhombus tilings is

Equality of flip-Hamming distances. The flip-Hamming distances in this case should be equal.