Tromino tilings of domino-deficient rectangles

Abstract

We consider tromino tilings of m × n domino-deficient rectangles, where 3|(mn − 2) and m, n ≥ 0, and characterize all cases of domino removal that admit such tilings, thereby settling the open problem posed by J. M. Ash and S. Golomb in [6]. Based on this characterization, we design a procedure for constructing such a tiling if one exists. We also consider the problem of counting such tilings and derive the exact formula for the number of tilings for 2 × (3t + 1) rectangles, the exact generating function for 4 × (3t + 2) rectangles, where t ≥ 0, and an upper bound on the number of tromino tilings for m × n domino-deficient rectangles. We also consider general 2-deficiency in n × 4 rectangles, where n ≥ 8, and characterize all pairs of squares which do not permit a tromino tiling.

Tetromino tilings on the Tetris board*

“…When confronted with polyominoes of larger sizes, the pursuit of exact enumeration formulas is seldom undertaken due to the intricate nature of the problem [11][12][13][14]. In this paper, our investigation centers specifically on tetrominos, which represent polyominoes composed of four unit squares.…”