Domino tilings of aztec diamonds and squares

The article gives several new combinatorial proofs of well-known theorems about domino tilings of aztec diamonds and squares. We prove also generalizations of these theorems for the generating polynomials of some statistics of tilings. Some results here are new, for example, we describe how to calculate the rank of a tiling using special weights of edges on the square grid.
The text is self-sufficient, we give numerous illustrations to all combinatorial constructions. Bibl. – 17 titles.