Geodesic
Enumeration of minimal 3D polyominoes inscribed in a rectangular prism
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011).

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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. 
@article{DMTCS_2011_special_260_a35,
     author = {Goupil, Alain and Cloutier, Hugo},
     title = {Enumeration of minimal {3D} polyominoes inscribed in a rectangular prism},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)},
     year = {2011},
     doi = {10.46298/dmtcs.2922},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2922/}
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Goupil, Alain; Cloutier, Hugo. Enumeration of minimal 3D polyominoes inscribed in a rectangular prism. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011). doi : 10.46298/dmtcs.2922. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2922/

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