Straight polyomino tilings of rectangles and special rim-hook tableaux

TL;DR

This paper derives explicit rational generating functions for weighted tilings of 2k by n rectangles with straight k by 1 tiles, using algebraic combinatorics and rim-hook tableaux.

Contribution

It introduces a novel approach combining fault line decomposition and Hadamard products to analyze tilings via rim-hook tableaux.

Findings

Explicit rational generating functions for tilings are obtained.

The approach connects tiling enumeration with algebraic combinatorics tools.

Graham's theorem on fault-free tilings is key to the analysis.

Abstract

We derive explicit rational generating functions for weighted tilings of $2k\times n$ rectangles by straight $k\times 1$ tiles. Our approach combines a decomposition by fault lines with a Hadamard-product framework. Tools from algebraic combinatorics are used together with a theorem of Klivans and Reiner on Schur expansions of plethystic compositions of elementary symmetric functions. This translates the tiling problem into a combinatorial framework via special rim-hook tableaux. On the tiling side, Graham's theorem on fault-free tilings provides the key input needed to complete the analysis.