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EXAMPLE
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Triangle begins:
1;
2;
3, 1;
4, 2;
5, 2;
6, 3, 1;
7, 3, 2;
8, 4, 3;
9, 4, 2;
10, 5, 3, 1;
11, 5, 4, 2;
12, 6, 3, 3;
13, 6, 4, 4;
14, 7, 5, 2;
15, 7, 4, 3, 1;
16, 8, 5, 4, 2;
17, 8, 6, 5, 3;
18, 9, 5, 3, 4;
19, 9, 6, 4, 5;
20, 10, 7, 5, 2;
21, 10, 6, 6, 3, 1;
22, 11, 7, 4, 4, 2;
23, 11, 8, 5, 5, 3;
24, 12, 7, 6, 6, 4;
25, 12, 8, 7, 3, 5;
26, 13, 9, 5, 4, 6;
27, 13, 8, 6, 5, 2;
28, 14, 9, 7, 6, 3, 1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts:
. ------------------------------------------------------------------------
Fig: A B C D E F G
. ------------------------------------------------------------------------
. n: 1 2 3 4 5 6 7
Row ------------------------------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; |
3 | | | [3],[1];| 3, 1;| 3, 1; | 3, 1; | 3, 1; |
4 | | | 4 ,[2];| [4], 2;| 4, 2; | 4, 2; | 4, 2; |
5 | | | | | [5],[2]; | 5, 2; | 5, 2; |
6 | | | | | 6, [3], 3;| [6], 3, [1];| 6, 3, 1;|
7 | | | | | | 7, 3, [2];| [7],[3], 2;|
8 | | | | | | 8, 4, [3];| 8, [4], 3;|
. ------------------------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts (but with the parts in increasing order) are [6] and [1, 2, 3]. These partitions have 1 and 3 consecutive parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 6.
.
Figures H..K show the location (in the columns of the table) of the partitions of 8..11 (respectively) into consecutive parts:
. --------------------------------------------------------------------
Fig: H I J K
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. n: 8 9 10 11
Row --------------------------------------------------------------------
1 | 1; | 1; | 1; | 1; |
1 | 2; | 2; | 2; | 2; |
3 | 3, 1; | 3, 1; | 3, 1; | 3, 1; |
4 | 4, 2; | 4, 2; | 4, 2; | 4, 2; |
5 | 5, 2; | 5, 2; | 5, 2; | 5, 2; |
6 | 6, 3, 3;| 6, 3, 1; | 6, 3, 1; | 6, 3, 1; |
7 | 7, 3, 2;| 7, 3, 2; | 7, 3, 2; | 7, 3, 2; |
8 | [8], 4, 1;| 8, 4, 3; | 8, 4, 3; | 8, 4, 3; |
9 | | [9],[4],[2]; | 9, 4, 2; | 9, 4, 2; |
10 | | 10, [5],[3], 1;| [10], 5, 3, [1];| 10, 5, 3, 1;|
11 | | 11, 5, [4], 2;| 11, 5, 4, [2];| [11],[5], 4, 2;|
12 | | | 12, 6, 3, [3];| 12, [6], 3, 3;|
13 | | | 13, 6, 4, [4];| 13, 6, 4, 4;|
. --------------------------------------------------------------------
Figure J: For n = 10 the partitions of 10 into consecutive parts (but with the parts in increasing order) are [10] and [1, 2, 3, 4]. These partitions have 1 and 4 consecutive parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 4 of this table, starting at the row 10.
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Illustration of initial terms arranged into the diagram of the triangle A237591:
. _
. _|1|
. _|2 _|
. _|3 |1|
. _|4 _|2|
. _|5 |2 _|
. _|6 _|3|1|
. _|7 |3 |2|
. _|8 _|4 _|3|
. _|9 |4 |2 _|
. _|10 _|5 |3|1|
. _|11 |5 _|4|2|
. _|12 _|6 |3 |3|
. _|13 |6 |4 _|4|
. _|14 _|7 _|5|2 _|
. _|15 |7 |4 |3|1|
. _|16 _|8 |5 |4|2|
. _|17 |8 _|6 _|5|3|
. _|18 _|9 |5 |3 |4|
. _|19 |9 |6 |4 _|5|
. _|20 _|10 _|7 |5|2 _|
. _|21 |10 |6 _|6|3|1|
. _|22 _|11 |7 |4 |4|2|
. _|23 |11 _|8 |5 |5|3|
. _|24 _|12 |7 |6 _|6|4|
. _|25 |12 |8 _|7|3 |5|
. _|26 _|13 _|9 |5 |4 _|6|
. _|27 |13 |8 |6 |5|2 _|
. |28 |14 |9 |7 |6|3|1|
...
The number of horizontal line segments in the n-th row of the diagram equals A001227(n), the number of partitions of n into consecutive parts.
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The connection (described step by step) between the triangle of A299765 and the above geometric diagram is as follows:
.
[1]; [1];
[2]; [2];
[3], [2, 1]; [3], [2, 1];
[4]; [4];
[5], [3, 2]; [5], [3, 2];
[6], [3, 2, 1]; [6], [3, 2, 1];
[7], [4, 3]; [7], [4, 3];
[8]; [8];
[9], [5, 4], [4, 3, 2]; [9], [5, 4], [4, 3, 2];
.
Figure 1. Figure 2.
.
We start with the irregular Then we write the same triangle
triangle of A299765 in which but ordered in columns where the
row n lists the partitions column k lists the partitions of
of n into consecutive parts. n into k consecutive parts.
.
. _ _
1| |1
_ _
2| |2
_ _ _ _ _
3| 2,1| |3 |1
_ _ |2
4| |4
_ _ _ _ _
5| 3,2| |5 |2
_ _ _ _ _ |3 _
6| 3,2,1| |6 |1
_ _ _ _ _ |2
7| 4,3| |7 |3 |3
_ _ |4
8| |8
_ _ _ _ _ _ _ _ _
9| 5,4| 4,3,2| |9 |4 |2
|5 |3
|4
.
Figure 3. Figure 4.
.
Then we draw to the right of Then we rotate each sub-diagram
each partition a vertical 90 degrees counterclockwise.
toothpick and above each part Every horizontal toothpick represents
we draw a horizontal toothpick. the existence of that partition.
. The number of vertical toothpicks
. equals the number of parts.
.
. _ _
_|1 _|1
_|2 _ _|2 _
_|3 |1 _|3 |1
_|4 _|2 _|4 _|2
_|5 |2 _ _|5 |2 _
_|6 _|3|1 _|6 _|3|1
_|7 |3 |2 _|7 |3 |2
_|8 _|4 _|3 _|8 _|4 _|3
|9 |4 |2 |9 |4 |2
|5 |3
|4
.
Figure 5. Figure 6.
.
Then we join the sub-diagrams Finally we erase the parts that
forming staircases (or zig-zag are beyond a certain level (in
paths) that represent the this case beyond the 9th level)
partitions that have the same to make the diagram more standard.
number of parts.
.
The numbers in the k-th staircase (from left to right) are the elements of the k-th column of the triangular array.
There is an infinite family of this kind of triangles, which are related to polygonal numbers and partitions into consecutive parts that differ by d. For more information see the theorems in A285914 and A303300.
Note that if we take two images of the diagram mirroring each other, with the y-axis in the middle of them, then a new diagram is formed, which is symmetric and represents the sequence A237593 as an isosceles triangle. Then if we fold each level (or row) of that isosceles triangle we essentially obtain the structure of the pyramid described in A245092 whose terraces at the n-th level have a total area equal to sigma(n) = A000203(n). (End)
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