Enumeration of minimal 3D polyominoes inscribed in a rectangular prism
نویسندگان
چکیده
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×k×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. Résumé. Nous considérons la famille des polyominos 3D de volume minimal inscrits dans un prisme rectangulaire. Ces objets sont des polyominos et sont donc des ensembles connexes de cubes unitaires. De plus ils sont inscrits dans un prisme rectangulaire de format b × k × h donné et ont un volume minimal égal à b + k + h − 2. Ces polyominos généralisent le concept de polyomino 2D étudié dans un travail précédent. Nous construisons des séries génératrices rationnelles de polyominos 3D minimaux et nous obtenons des formules exactes et des récurrences pour des sous-familles de ces polyominos.
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