On Propp's Problem 12: The Sum of the Entries of the Inverse Kasteleyn–Percus Matrix of the Aztec Diamond

Let \(K_n\) be the Kasteleyn–Percus matrix of the Aztec diamond of order \(n\): the adjacency matrix between its black and its white squares in which the entry of every vertical domino whose white square is on top is replaced by \(-1\). In Problem 12 of his list of open problems in the enumeration of matchings (1996, revised 1998; the sign convention is fixed in the version of 1999, where the problem is still open), J. Propp asked for a proof that the sum of the entries of \(K_n^{-1}\) is \(\frac12(n-1)(n+3)-2^{n-1}+2\), a formula he had checked for \(n\le8\). We prove it for every \(n\ge1\). The matrix \(K_n\) maps the constant function \(\frac12\) on the white squares to the constant \(1\), except on one edge of the diamond; in coordinates rotated by \(45^\circ\) the remaining equation has an explicit solution whose values are the Krawtchouk coefficients \(-\frac12[\xi^m](1-\xi)^k(1+\xi)^{n-k}\), and two transpositions reduce the entry sum to \(\frac{n(n+1)}2+\frac n2-\frac{2^n-1}2=\frac{(n+1)^2}2-2^{n-1}\). The argument also gives every row sum and every column sum of \(K_n^{-1}\) as an explicit Krawtchouk coefficient. The problem and the formula are Propp's; the matrix and the determinant method are due to Kasteleyn and Percus; the Aztec diamond and the number \(2^{n(n+1)/2}\) of its domino tilings to Elkies, Kuperberg, Larsen and Propp; explicit formulas for the individual entries of the inverse, as sums of products of Krawtchouk coefficients, to Helfgott (1998, posted 2000), whose coupling function differs from the inverse by an explicit sign factor, extended to different weights for horizontal and vertical dominoes by Chhita, Johansson and Young. These works do not compute the entry sum or the row sums. What the note adds is an elementary proof of Propp's formula and the row and column sums. Computations show that the entry sum depends on the choice of the Kasteleyn signs, so that the convention of the 1999 text is essential. This is an unrefereed note.