For n = 5 consider the partitions of 5 in colexicographic order (as shown in the 5th row of the triangle A211992) and its associated diagram of regions as shown below:
. Regions Minimalist
. Partitions of 5 diagram version
. _ _ _ _ _
. 1, 1, 1, 1, 1 |_| | | | | _| | | | |
. 2, 1, 1, 1 |_ _| | | | _ _| | | |
. 3, 1, 1 |_ _ _| | | _ _ _| | |
. 2, 2, 1 |_ _| | | _ _| | |
. 4, 1 |_ _ _ _| | _ _ _ _| |
. 3, 2 |_ _ _| | _ _| |
. 5 |_ _ _ _ _| _ _ _ _ _|
.
Then consider the following table which contains the Ferrers boards of the partitions of 5 and the diagram of every region of the set of partitions of 5:
-------------------------------------------------------------------------
| Partitions | | | Regions | | |
| of 5 | Ferrers | Peri- | of 5 | Region | Peri- |
-------------------------------------------------------------------------
| _ | _ |
| 1 |_| | 1 |_| 4 |
| 1 |_| | _ |
| 1 |_| | 1 _|_| |
| 1 |_| | 2 |_|_| 8 |
| 1 |_| 12 | _ |
| _ _ | 1 |_| |
| 2 |_|_| | 1 _ _|_| |
| 1 |_| | 3 |_|_|_| 12 |
| 1 |_| | _ _ |
| 1 |_| 12 | 2 |_|_| 6 |
| _ _ _ | _ |
| 3 |_|_|_| | 1 |_| |
| 1 |_| | 1 |_| |
| 1 |_| 12 | 1 _|_| |
| _ _ | 2 _ _|_|_| |
| 2 |_|_| | 4 |_|_|_|_| 18 |
| 2 |_|_| | _ _ _ |
| 1 |_| 10 | 3 |_|_|_| 8 |
| _ _ _ _ | _ |
| 4 |_|_|_|_| | 1 |_| |
| 1 |_| 12 | 1 |_| |
| _ _ _ | 1 |_| |
| 3 |_|_|_| | 1 |_| |
| 2 |_|_| 10 | 1 _|_| |
| _ _ _ _ _ | 2 _ _ _|_|_| |
| 6 |_|_|_|_|_| 12 | 5 |_|_|_|_|_| 24 |
| | |
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| Sum of perimeters: 80 <-- equals --> 80 |
-------------------------------------------------------------------------
The sum of the perimeters of the Ferrers boards is 12 + 12 + 12 + 10 + 12 + 10 + 12 = 80, so a(5) = 80.
On the other hand, the sum of the perimeters of the diagrams of regions is 4 + 8 + 12 + 6 + 18 + 8 + 24 = 80, equaling the sum of the perimeters of the Ferrers boards.
.
Illustration of first six polygons of an infinite diagram constructed with the boundary segments of the minimalist diagram of regions and its mirror (note that the diagram looks like reflections on a mountain lake):
11............................................................
. /\
. / \
. / \
7................................... / \
. /\ / \
5..................... / \ /\/ \
. /\ / \ /\ / \
3........... / \ / \ / \/ \
2....... /\ / \ /\/ \ / \
1... /\ / \ /\/ \ / \ /\/ \
0 /\/ \/ \/ \/ \/ \
. \/\ /\ /\ /\ /\ /
. \/ \ / \/\ / \ / \/\ /
. \/ \ / \/\ / \ /
. \ / \ / \ /\ /
. \/ \ / \/ \ /
. \ / \/\ /
. \/ \ /
. \ /
. \ /
. \ /
. \/
n:
. 0 1 2 3 4 5 6
Perimeter of the n-th polygon:
. 0 4 8 12 24 32 60
a(n) is the sum of the perimeters of the first n polygons:
. 0 4 12 24 48 80 140
.
For n = 5, the sum of the perimeters of the first five polygons is 4 + 8 + 12 + 24 + 32 = 80, so a(5) = 80.
For n = 6, the sum of the perimeters of the first six polygons is 4 + 8 + 12 + 24 + 32 + 60 = 140, so a(6) = 140.
For another version of the above diagram see A228109.
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