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A365545
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Triangle read by rows where T(n,k) is the number of strict integer partitions of n with exactly k distinct non-subset-sums.
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10
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1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 3, 0, 1, 0, 0, 1, 1, 0, 0, 3, 0, 1, 0, 0, 0, 3, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 5, 0, 0, 0, 5, 0, 1, 0, 2, 0, 0, 0, 0, 5, 2, 0, 0, 5, 0, 1, 0
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OFFSET
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0,18
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COMMENTS
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For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.
Is column k = n - 7 given by A325695?
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LINKS
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EXAMPLE
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Triangle begins:
1
1 0
0 1 0
1 0 1 0
0 1 0 1 0
0 0 2 0 1 0
1 0 0 2 0 1 0
1 0 0 0 3 0 1 0
0 1 1 0 0 3 0 1 0
0 0 3 0 0 0 4 0 1 0
1 0 0 2 2 0 0 4 0 1 0
1 0 0 0 5 0 0 0 5 0 1 0
2 0 0 0 0 5 2 0 0 5 0 1 0
2 0 1 0 0 0 8 0 0 0 6 0 1 0
1 1 3 0 0 0 0 7 3 0 0 6 0 1 0
2 0 4 0 1 0 0 0 12 0 0 0 7 0 1 0
1 1 2 2 3 1 0 0 0 11 3 0 0 7 0 1 0
2 0 3 0 7 0 1 0 0 0 16 0 0 0 8 0 1 0
3 0 0 2 6 3 3 1 0 0 0 15 4 0 0 8 0 1 0
Row n = 12: counts the following partitions:
(6,3,2,1) . . . . (9,2,1) (6,5,1) . . (11,1) . (12) .
(5,4,2,1) (8,3,1) (6,4,2) (10,2)
(7,4,1) (9,3)
(7,3,2) (8,4)
(5,4,3) (7,5)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Complement[Range[n], Total/@Subsets[#]]]==k&]], {n, 0, 10}, {k, 0, n}]
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CROSSREFS
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The complement (positive subset-sums) is also A365545 with rows reversed.
A046663 counts partitions without a subset summing to k, strict A365663.
A364350 counts combination-free strict partitions, complement A364839.
A365543 counts partitions with a submultiset summing to k, strict A365661.
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KEYWORD
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AUTHOR
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STATUS
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approved
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