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A367405
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Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.
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9
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1, 0, 1, 0, 0, 2, 1, 1, 1, 2, 1, 0, 1, 1, 3, 1, 1, 1, 1, 2, 3, 1, 1, 1, 2, 2, 2, 4, 2, 2, 3, 2, 3, 2, 3, 4, 2, 2, 3, 2, 3, 3, 3, 3, 5, 3, 2, 4, 3, 4, 4, 5, 3, 4, 5, 3, 3, 5, 4, 4, 5, 5, 5, 4, 4, 6, 4, 3, 6, 5, 6, 5, 7, 5, 7, 4, 5, 6, 5, 5, 7, 7, 8, 7, 8, 8, 7, 7, 5, 5, 7
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OFFSET
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3,6
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 0 2
1 1 1 2
1 0 1 1 3
1 1 1 1 2 3
1 1 1 2 2 2 4
2 2 3 2 3 2 3 4
2 2 3 2 3 3 3 3 5
3 2 4 3 4 4 5 3 4 5
3 3 5 4 4 5 5 5 4 4 6
4 3 6 5 6 5 7 5 7 4 5 6
5 5 7 7 8 7 8 8 7 7 5 5 7
6 5 9 8 10 7 10 9 10 7 9 5 6 7
7 7 10 10 12 11 11 11 12 10 9 9 6 6 8
9 7 13 11 15 12 13 13 15 13 13 9 11 6 7 8
Row n = 9 counts the following strict partitions:
(6,2,1) (5,3,1) (4,3,2) (5,3,1) (6,2,1) (6,2,1) (8,1)
(4,3,2) (4,3,2) (5,3,1) (7,2)
(6,3)
(5,4)
Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
A21 931 841 751 652 751 841 931 A21 A21 C1
7321 7321 832 742 643 7321 742 832 832 931 B2
6421 5431 7321 6421 6421 652 7321 7321 742 841 A3
6421 5431 5431 6421 643 643 652 751 94
5431 5431 5431 6421 85
76
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#, {2}], k]&]], {n, 3, 10}, {k, 3, n}]
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CROSSREFS
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For subsets instead of partitions we have A365541, non-binary A365381.
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KEYWORD
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AUTHOR
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STATUS
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approved
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