Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

Abstract : We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size $b\times k \times h$ and of minimal volume equal to $b+k+h-2$. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structure and elementary combinatorial principles, we construct rational generating functions of minimal 3D polyominoes. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.
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https://hal.inria.fr/hal-01215105
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Alain Goupil, Hugo Cloutier. Enumeration of minimal 3D polyominoes inscribed in a rectangular prism. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.423-434. ⟨hal-01215105⟩

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