Enumeration of polyominoes inscribed in a rectangle

Enumeration of polyominoes inscribed in a rectangle

Enumeration of minimal 3D polyominoes inscribed in a rectangular prism

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 structur...

Tiling a Rectangle with Polyominoes

A polycube in dimension d is a finite union of unit d-cubes whose vertices are on knots of the lattice Zd . We show that, for each family of polycubes E, there exists a finite set F of bricks (parallelepiped rectangles) such that the bricks which can be tiled by E are exactly the bricks which can be tiled by F . Consequently, if we know the set F , then we have an algorithm to decide in polynom...

Enumeration of inscribed polyominos

We introduce a new family of polyominos that are inscribed in a rectangle of given size for which we establish a number of exact formulas and generating functions. In particular, we study polyominos inscribed in a rectangle with minimum area and minimum area plus one. These results are then used for the enumeration of lattice trees inscribed in a rectangle with minimum area plus one. Résumé. No...

Enumeration of generalized polyominoes

As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular k-gons. For n ≤ 4 we determine formulas for the number ak(n) of generalized polyominoes consisting of n regular k-gons. Additionally we give a table of the numbers ak(n) for small k and n obtained by computer enumeration. We finish with some open problems for k-polyominoes.