Tromino tilings of Domino-Deficient Rectangles

TL;DR

This paper characterizes when tromino tilings are possible in domino-deficient rectangles, provides construction methods, exact formulas for specific cases, and bounds on the number of tilings, advancing understanding of tiling combinatorics.

Contribution

It offers a complete characterization of domino removal cases allowing tromino tilings and develops formulas and bounds for counting such tilings.

Findings

Characterized all domino removal cases permitting tromino tilings.

Derived exact formulas for tilings of specific rectangle sizes.

Provided bounds and characterizations for general 2-deficiency cases.

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.