OFFSET
1,2
COMMENTS
The sequence lists the partitions of all positive integers. Each row of the irregular array is a partition of j.
At stage 1, we start with 1.
At stage j > 1, we write the partitions of j using the following rules:
First, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, starting with the partition that contains the part of size j.
Second, we copy from this array the partitions of j-1 in descending order, as a mirror image, starting with the partition that contains the part of size j-2 together with the part of size 1. At the end of each new row, we added a part of size 1.
EXAMPLE
A table of partitions.
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. Expanded Geometric
Partitions arrangement model
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1; 1; |*|
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2; . 2; |* *|
1,1; 1,1; |o|*|
--------------------------------------------
3; . . 3; |* * *|
1,1,1; 1,1,1; |o|o|*|
2,1; . 2,1; |o o|*|
--------------------------------------------
4; . . . 4; |* * * *|
2,2; . 2,. 2; |* *|* *|
2,1,1; . 2,1,1; |o o|o|*|
1,1,1,1; 1,1,1,1; |o|o|o|*|
3,1; . . 3,1; |o o o|*|
--------------------------------------------
5; . . . . 5; |* * * * *|
3,2; . . 3,. 2; |* * *|* *|
3,1,1; . . 3,1,1; |o o o|o|*|
1,1,1,1,1; 1,1,1,1,1; |o|o|o|o|*|
2,1,1,1; . 2,1,1,1; |o o|o|o|*|
2,2,1; . 2,. 2,1; |o o|o o|*|
4,1; . . . 4,1; |o o o o|*|
--------------------------------------------
6; . . . . . 6; |* * * * * *|
3,3; . . 3,. . 3; |* * *|* * *|
4,2; . . . 4,. 2; |* * * *|* *|
2,2,2; . 2,. 2,. 2; |* *|* *|* *|
4,1,1; . . . 4,1,1; |o o o o|o|*|
2,2,1,1; . 2,. 2,1,1; |o o|o o|o|*|
2,1,1,1,1; . 2,1,1,1,1; |o o|o|o|o|*|
1,1,1,1,1,1; 1,1,1,1,1,1; |o|o|o|o|o|*|
3,1,1,1; . . 3,1,1,1; |o o o|o|o|*|
3,2,1; . . 3,. 2,1; |o o o|o o|*|
5,1; . . . . 5,1; |o o o o o|*|
--------------------------------------------
Note that * is a unitary element of every part of the last section of j.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 18 2012
STATUS
approved