2004/06/15 by James Propp
Mathematics · #math.CO #msc:05A99
published as Advances in Applied Mathematics, Volume 34, Issue 4, May 2005, pp. 871-879 · 4 pages; to appear in a special issue of Advances in Applied Mathematics honoring David P. Robbins
arxiv created 2004/06/15 · arxiv updated 2009/12/01
Consider the 2n-by-2n matrix M=(mi,j)i,j=12n with mi,j = 1 for i,j satisfying |2i-2n-1|+|2j-2n-1| ≤ 2n and mi,j = 0 for all other i,j, consisting of a central diamond of 1's surrounded by 0's. When n ≥ 4, the λ-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 λ-determinant is well-defined and is a polynomial in λ and t. The limit of this polynomial as t → 0 is a polynomial in λ whose value at λ=1 is the number of domino tilings of a 2n-by-2n square.