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A365832
Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.
8
1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 0, 1, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 0, 1, 0, 1, 0, 5, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 5, 0, 0, 2, 5, 0, 0, 0, 0, 2
OFFSET
0,19
EXAMPLE
The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
Triangle begins:
1
0 1
0 1 0
0 1 0 1
0 1 0 1 0
0 1 0 2 0 0
0 1 0 2 0 0 1
0 1 0 3 0 0 0 1
0 1 0 3 0 0 1 1 0
0 1 0 4 0 0 0 3 0 0
0 1 0 4 0 0 2 2 0 0 1
0 1 0 5 0 0 0 5 0 0 0 1
0 1 0 5 0 0 2 5 0 0 0 0 2
0 1 0 6 0 0 0 8 0 0 0 1 0 2
0 1 0 6 0 0 3 7 0 0 0 0 3 1 1
0 1 0 7 0 0 0 12 0 0 0 1 0 4 0 2
0 1 0 7 0 0 3 11 0 0 0 1 3 2 2 1 1
0 1 0 8 0 0 0 16 0 0 0 1 0 7 0 3 0 2
0 1 0 8 0 0 4 15 0 0 0 1 3 3 6 2 0 0 3
0 1 0 9 0 0 0 21 0 0 0 2 0 9 0 7 0 1 0 4
0 1 0 9 0 0 4 20 0 0 1 0 4 8 5 5 0 0 2 0 5
Row n = 14 counts the following partitions (A..E = 10..14):
(E) . (D1) . . (761) (B21) . . . . (6521) (8321) (7421)
(C2) (752) (A31) (6431)
(B3) (743) (941) (5432)
(A4) (932)
(95) (851)
(86) (842)
(653)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Length[Union[Total/@Rest[Subsets[#]]]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Row sums are A000009.
Rightmost column n = k is A188431, non-strict A126796.
The one-based weighted row sums are A284640.
The corresponding rank statistic is A299701.
The non-strict version is A365658.
Central column n = 2k in the non-strict case is A365660.
Reverse-weighted row-sums are A365922, non-strict A276024.
A000041 counts integer partitions.
A000124 counts distinct sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Sequence in context: A135767 A208575 A355037 * A070203 A070201 A070138
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 28 2023
STATUS
approved