Augmented Aztec bipyramid and dicube tilings

Sunmook Choi, Sangyop Lee, Seungsang Oh

Abstract: We consider the enumeration of dicube tilings, where each tiling represents a three-dimensional tessellation of a polycube using dicubes. While the enumeration of domino tilings of polycubes like the Aztec diamond and the augmented Aztec diamond is well studied, we focus on the three-dimensional analogue, the augmented Aztec bipyramid. This polycube consists of unit cubes and resembles a Platonic octahedron. In this paper, we find a bijection between dicube tilings of the augmented Aztec bipyramid and three-dimensional Delannoy paths, and use this correspondence to determine the number of dicube tilings of the augmented Aztec bipyramid.