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A211999
A list of ordered partitions of the positive integers in which the shells of each integer are assembled by their tails.
9
1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 4, 4, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 2, 5, 5, 1, 3, 2, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 2, 4, 2, 3, 3, 6, 6, 1, 3, 3, 1, 4, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 5, 1, 1, 3, 2, 2, 5, 2, 4, 3, 7
OFFSET
1,4
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 copy the last A000041(j-1) rows of the array in descending order, as a mirror image, starting with the row that contains the part of size j-1. At the end of each new row, we added a part of size 1.
Second, we write the partitions of j that do not contain 1 as a part, in reverse-lexicographic order, such that the last row (or partition of j) is j.
Note that the table can be partially folded. In this case we have a three-dimensional structure in which each column contains parts of the same size (see example). Also the table can be completely folded, therefore stacked parts have the same size.
EXAMPLE
A table of partitions.
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. Expanded Geometric Side view of the
Partitions version model folded table
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1; 1; |*| /
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1,1; 1,1; |o|*| \
2; . 2; |* *| \
---------------------------------------------------------
2,1; . 2,1; |o o|*| /
1,1,1; 1,1,1; |o|o|*| /
3; . . 3; |* * *| /
---------------------------------------------------------
3,1; . . 3,1; |o o o|*| \
1,1,1,1; 1,1,1,1; |o|o|o|*| \
2,1,1; . 2,1,1; |o o|o|*| \
2,2; . 2,. 2; |* *|* *| \
4; . . . 4; |* * * *| \
---------------------------------------------------------
4,1; . . . 4,1; |o o o o|*| /
2,2,1; . 2,. 2,1; |o o|o o|*| /
2,1,1,1; . 2,1,1,1; |o o|o|o|*| /
1,1,1,1,1; 1,1,1,1,1; |o|o|o|o|*| /
3,1,1; . . 3,1,1; |o o o|o|*| /
3,2; . . 3,. 2; |* * *|* *| /
5; . . . . 5; |* * * * *| /
---------------------------------------------------------
5,1; . . . . 5,1; |o o o o o|*| \
3,2,1; . . 3,. 2,1; |o o o|o o|*| \
3,1,1,1; . . 3,1,1,1; |o o o|o|o|*| \
1,1,1,1,1,1; 1,1,1,1,1,1; |o|o|o|o|o|*| \
2,1,1,1,1; . 2,1,1,1,1; |o o|o|o|o|*| \
2,2,1,1; . 2,. 2,1,1; |o o|o o|o|*| \
4,1,1; . . . 4,1,1; |o o o o|o|*| \
2,2,2; . 2, .2,. 2; |* *|* *|* *| \
4,2; . . . 4,. 2; |* * * *|* *| \
3,3; . . 3,. . 3; |* * *|* * *| \
6; . . . . . 6; |* * * * * *| \
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Note that * is a unitary element of every part of the last section of j.
CROSSREFS
Rows sums give A036042, n>=1.
Other versions are A211983, A211984, A211989. See also A026792, A211992-A211994. Spiral arrangements are A211985-A211988, A211995-A211998.
Sequence in context: A099245 A185331 A206474 * A175025 A076899 A152905
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Aug 14 2012
STATUS
approved