2016/01/04 by Kenneth Gill, Gill, Kenneth, Viorel Niţică +1
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Cellular Automata and Applications #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1601.00572
openalex publication_date 2016/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Tn be the set of ribbon L-shaped n-ominoes for some n≥ 4 even, and let Tn+ be Tn with an extra 2× 2 square. We investigate signed tilings of rectangles by Tn and Tn+. We show that a rectangle has a signed tiling by Tn if and only if both sides of the rectangle are even and one of them is divisible by n, or if one of the sides is odd and the other side is divisible by n ((n)/(2)-2 ). We also show that a rectangle has a signed tiling by Tn+, n≥ 6 even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by n ((n)/(2)-2 ). Our proofs are based on the exhibition of explicit Gröbner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph V.~Nitica, Every tiling of the first quadrant by ribbon L n-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, \em 5, (2015) 11--25, cannot be obtained from coloring invariants.