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A365658
Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.
35
1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55
OFFSET
1,6
COMMENTS
Conjecture: Positions of strictly positive rows are given by A048166.
EXAMPLE
Triangle begins:
1
1 1
1 0 2
1 1 1 2
1 0 2 0 4
1 1 3 0 1 5
1 0 3 0 3 0 8
1 1 3 2 2 1 2 10
1 0 5 0 3 0 5 0 16
1 1 4 0 6 2 4 2 2 20
1 0 5 0 5 0 8 0 6 0 31
1 1 6 2 3 6 6 1 4 4 4 39
1 0 6 0 6 0 12 0 8 0 13 0 55
1 1 6 0 6 3 16 3 5 3 7 8 5 71
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Rest[Subsets[#]]]]==k&]], {n, 10}, {k, n}]
CROSSREFS
Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Sequence in context: A112400 A316523 A219185 * A116861 A340032 A327785
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 16 2023
STATUS
approved