Straight polyomino tilings of rectangles and special rim-hook tableaux

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.