Some 2-adic conjectures concerning polyomino tilings of Aztec diamonds

Abstract

For various sets of tiles, we count the ways to tile an Aztec diamond of order $n$ using tiles from that set. The resulting function $f(n)$ often has interesting behavior when one looks at $n$ and $f(n)$ modulo powers of 2.

Note: The proof given in the second-to-last paragraph of section 5 is incorrect. See the published version of the paper for a correct proof