Lambda-determinants and domino-tilings

Abstract

Consider the $2n$-by-$2n$ matrix $M=(m_{i,j}{)}_{i,j=1}^{2n}$ with $m_{i,j}=1$ for $i,j$ satisfying $|2i-2n-1|+|2j-2n-1|\le 2n$ and $m_{i,j}=0$ for all other $i,j$, consisting of a central diamond of 1's surrounded by 0's. When $n\ge 4$, the $\hat{I}»$-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 $\hat{I}»$-determinant is well-defined and is a polynomial in $\hat{I}»$ and $t$. The limit of this polynomial as $t\to 0$ is a polynomial in $\hat{I}»$ whose value at $\hat{I}»=1$ is the number of domino tilings of a $2n$-by-$2n$ square.

4 pages; to appear in a special issue of Advances in Applied Mathematics honoring David P. Robbins