  Consider the $2n$-by-$2n$ matrix $M=(m_{i,j})_{i,j=1}^{2n}$ with $m_{i,j} =
1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1| \leq 2n$ and $m_{i,j} = 0$ for all
other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n
\geq 4$, the $\lambda$-determinant of the matrix $M$ (as introduced by Robbins
and Rumsey) is not well-defined. However, if we replace the 0's by $t$'s, we
get a matrix whose $\lambda$-determinant is well-defined and is a polynomial in
$\lambda$ and $t$. The limit of this polynomial as $t \to 0$ is a polynomial in
$\lambda$ whose value at $\lambda=1$ is the number of domino tilings of a
$2n$-by-$2n$ square.
