5 problems

Shift-invariance conjecture. If $\\{\alpha,\beta\\}$ is not an inversion of $w$, then the three probability ratios satisfy

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…

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…