Lambda-determinants and domino-tilings

Details Lambda-determinants and domino-tilings Lambda-determinants and domino-tilings

2004-06-15

arxiv.org/abs/math/0406301v1

Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, pp. 871-879

Mathematics

Combinatorics

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

Scientific paper

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.

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