Gröbner Lattice-Point Enumerators and Signed Tiling by k-in-line Polyominoes

Abstract

Conway and Lagarias observed that a triangular region T 2 (n) in a hexagonal lattice admits a signed tiling by 3-in-line polyominoes (tribones) if and only if n ∈ {3 2 d − 1, 3 2 d} d∈N . We apply the theory of Gröbner bases over integers to show that T 3 (n), a three dimensional lattice tetrahedron of edge-length n, admits a signed tiling by tribones if and only if n ∈ {3 3 d − 2, 3 3 d − 1, 3 3 d} d∈N . More generally we study Gröbner lattice-point enumerators of lattice polytopes and show that they are (modular) quasipolynomials in the case of k-in-line polyominoes. As an example of the "unusual cancelation phenomenon", arising only in signed tilings, we exhibit a configuration of 15 tribones in the 3-space such that exactly one lattice point is covered by an odd number of tiles.