3 problems

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…

Let $n=4m$, $0<a<1$, and take $x=(x_1,x_2)\in\mathtt{W}_{\varepsilon_1}$ and $y=(y_1,y_2)\in\mathtt{B}_{\varepsilon_2}$ with $\varepsilon_1,\varepsilon_2\in\\{0,1\\}$. Let…

Let $x \in \mathtt{W}_{\varepsilon_1}$ and $y \in \mathtt{B}_{\varepsilon_2}$ with $\varepsilon_1,\varepsilon_2 \in \\{0,1\\}$. Let $K_0(\alpha)$ and $K_1(\alpha)$ denote modified…