For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower". Both objects contains the same number of cubes (that property is also valid for n >= 1).
_ _ _ _ _ _ _
|_ _ _ _ | 7
|_ _ _ _|_ | 4 3
|_ _ _ | | 5 2
|_ _ _|_ _|_ | 3 2 2 _
|_ _ _ | | 6 1 1 | |
|_ _ _|_ | | 3 3 1 1 | |
|_ _ | | | 4 2 1 1 | |
|_ _|_ _|_ | | 2 2 2 1 1 _|_|
|_ _ _ | | | 5 1 1 1 1 | |
|_ _ _|_ | | | 3 2 1 1 1 1 _|_ _|
|_ _ | | | | 4 1 1 1 1 1 1 | | |
|_ _|_ | | | | 2 2 1 1 1 1 1 1 _|_|_ _|
|_ _ | | | | | 3 1 1 1 1 1 1 1 1 _| |_ _ _|
|_ | | | | | | 2 1 1 1 1 1 1 1 1 1 1 _ _|_ _|_ _ _|
|_|_|_|_|_|_|_| 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |_ _|_|_ _ _ _|
.
Figure 1. Figure 2. Figure 3. Figure 4.
Front view of the Partitions Position Lateral view
prism of partitions. of 7. of the 1's. of the tower.
.
.
_ _ _ _ _ _ _
| | | | | |_| 1
| | | |_|_ _| 2
| |_|_ |_ _| 3
|_ _ |_ _ _| 4
|_ |_ _ _| 5
| | 6
|_ _ _ _| 7
.
Figure 5.
Top view
of the tower.
.
Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is
A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*
A000041(7) = 1*7*15 = 105, equaling the volume of the tower that appears in the figures 4 and 5.
Figure 2 shows the partitions of 7 in accordance with the diagram.
Note that the shape and the area of the lateral view of the tower are the same as the shape and the area where the 1's are located in the diagram of partitions, see the figures 3 and 4. In this case the mentioned area equals
A000070(7-1) = 30.
The connection between these two objects is a representation of the correspondence divisor/part described in
A338156. See also
A336812.
