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A207383
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Triangle read by rows: T(n,k) is the sum of parts of size k in the last section of the set of partitions of n.
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16
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1, 1, 2, 2, 0, 3, 3, 4, 0, 4, 5, 2, 3, 0, 5, 7, 8, 6, 4, 0, 6, 11, 6, 6, 4, 5, 0, 7, 15, 16, 9, 12, 5, 6, 0, 8, 22, 14, 18, 8, 10, 6, 7, 0, 9, 30, 30, 18, 20, 15, 12, 7, 8, 0, 10, 42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11, 56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12
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OFFSET
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1,3
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COMMENTS
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For further properties of this triangle see also A182703.
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1;
1, 2;
2, 0, 3;
3, 4, 0, 4;
5, 2, 3, 0, 5;
7, 8, 6, 4, 0, 6;
11, 6, 6, 4, 5, 0, 7;
15, 16, 9, 12, 5, 6, 0, 8;
22, 14, 18, 8, 10, 6, 7, 0, 9;
30, 30, 18, 20, 15, 12, 7, 8, 0, 10;
42, 30, 30, 20, 20, 12, 14, 8, 9, 0, 11;
56, 54, 42, 40, 25, 30, 14, 16, 9, 10, 0, 12;
...
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7:
. _ _ _ _ _ _ _
. (7) (7) |_ _ _ _ |
. (4+3) (4+3) |_ _ _ _|_ |
. (5+2) (5+2) |_ _ _ | |
. (3+2+2) (3+2+2) |_ _ _|_ _|_ |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) | |
. (1) (1) |_|
. ----------------
. 19,8,5,3,2,1,1 --> Row 7 of triangle A207031
. |/|/|/|/|/|/|
. 11,3,2,1,1,0,1 --> Row 7 of triangle A182703
. * * * * * * *
. 1,2,3,4,5,6,7 --> Row 7 of triangle A002260
. = = = = = = =
. 11,6,6,4,5,0,7 --> Row 7 of this triangle
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Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So the 7th row of triangle is [11, 6, 6, 4, 5, 0, 7]. (End)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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