3D Polyominoes inscribed in a rectangular prism
Abstract
We introduce a family of 3D combinatorial objects that we define as minimal 3D polyominoes inscribed in a rectanglar prism. These objects are connected sets of unitary cubic cells inscribed in a given rectangular prism and of minimal volume under this condition. They extend the concept of 2D polyominoes inscribed in a rectangle defined in a previous work. Using their geometric structure and elementary combinatorial arguments, we construct generating functions of minimal 3D polyominoes in the form of rational functions. We also obtain a number of exact formulas and recurrences for sub-families of these polyominoes.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2010
- DOI:
- 10.48550/arXiv.1009.4859
- arXiv:
- arXiv:1009.4859
- Bibcode:
- 2010arXiv1009.4859A
- Keywords:
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- Mathematics - Combinatorics;
- 05A15
- E-Print:
- 15 pages