Generating functions for straight polyomino tilings of narrow rectangles

TL;DR

This paper derives explicit generating functions for tilings of narrow rectangles with fixed-width tiles, solving a long-standing open problem in combinatorics and extending results to three-dimensional tilings.

Contribution

The paper provides an explicit formula for the generating function when the rectangle's height is less than twice the tile width, advancing understanding of polyomino tilings.

Findings

Explicit generating function for m<2k case

Extension to 3D tilings with k×k×1 bricks

Addresses a long-standing open problem

Abstract

Let $m,k$ be fixed positive integers. Determining the generating function for the number of tilings of an $m\times n$ rectangle by $k\times 1$ rectangles is a long-standing open problem to which the answer is only known in certain special cases. We give an explicit formula for this generating function in the case where $m<2k$. This result is used to obtain the generating function for the number of tilings of an $m\times n \times k$ box with $k\times k\times 1$ bricks.