Tromino tilings of Domino-Deficient Rectangles

Abstract

We consider tromino tilings of $m\times n$ domino-deficient rectangles, where $3|(mn-2)$ and $m,n\geq0$, 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 \cite {marshall}. 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 $2\times(3t+1)$ rectangles, the exact generating function for $4\times(3t+2)$ rectangles, where $t\geq0$, and an upper bound on the number of tromino tilings for $m\times n$ domino-deficient rectangles. We also consider general 2-deficiency in $n\times4$ rectangles, where $n\geq8$, and characterize all pairs of squares which do not permit a tromino tiling.