Some 2-adic conjectures concerning polyomino tilings of Aztec diamonds

TL;DR

This paper investigates the behavior of tiling counts of Aztec diamonds with various tiles, focusing on the properties of the counting function modulo powers of 2, and explores related 2-adic conjectures.

Contribution

It introduces new conjectures linking tiling counts of Aztec diamonds to 2-adic properties and analyzes their behavior across different tile sets.

Findings

Counting functions exhibit interesting 2-adic properties.

Modular behavior of tiling counts reveals underlying mathematical structures.

New conjectures proposed relating tilings to 2-adic number theory.

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.