Symmetric Domino Tilings of Aztec Diamonds

In this paper, we give inductive sum formulas to calculate the number of diagonally symmetric, and diagonally & anti-diagonally symmetric domino tilings of Aztec Diamonds. As a byproduct, we also find such a formula for the unrestricted case as well. Our proofs rely on a new technique for counting the number of perfect matchings of graphs, proposed by the authors recently.