Masao ISHIKAWA: Domino tilings of Aztec rectangle with connected holes

It is famous that the number of domino tilings of an Aztec diamond is \(2\) to \(n+1\) choose \(2\). We study the number of domino tilings of an Aztec rectangle with even number of connected holes in a line and we obtain a formula which express the number of such domino tilings by a product of a similar power of 2, linear factors and a polynomial of the coordinates of the holes in a line. We will find a formula which expresses this polynomial as a determinant of terminating Gauss hypergeometric series and show that this polynomial possesses interesting properties. First we use the Lindstrom-Gessel-Viennot theorem to enumerate the domino tilings of an Aztec rectangle with connected holes and obtain a determinant whose entries are generalized large Schr\"oder numbers. Then we consider a more general determinant whose entries are the moments of the Laurent biorthogonal polynomials, which enable us to apply the Desnanot-Jacobi adjoint matrix theorem. This general determinant reduces to the case \(q=t=1\) in Kamioka's result if we have no hole, i.e., the Aztec diamond case. Then the evaluation of the determinant reduces to a quadratic relation of the above polynomials. This project is still a work in progress and we believe that we are very close to the complete proof. This is a joint work with Fumihiko Nakano, Taizo Sadahiro and Hiroyuki Tagawa.