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%I #5 Sep 30 2023 09:22:18
%S 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,
%T 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,
%U 0,5,0,0,0,5,0,0,0,1,0,1,0,5,0,0,2,5,0,0,0,0,2
%N Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.
%e The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 0
%e 0 1 0 1
%e 0 1 0 1 0
%e 0 1 0 2 0 0
%e 0 1 0 2 0 0 1
%e 0 1 0 3 0 0 0 1
%e 0 1 0 3 0 0 1 1 0
%e 0 1 0 4 0 0 0 3 0 0
%e 0 1 0 4 0 0 2 2 0 0 1
%e 0 1 0 5 0 0 0 5 0 0 0 1
%e 0 1 0 5 0 0 2 5 0 0 0 0 2
%e 0 1 0 6 0 0 0 8 0 0 0 1 0 2
%e 0 1 0 6 0 0 3 7 0 0 0 0 3 1 1
%e 0 1 0 7 0 0 0 12 0 0 0 1 0 4 0 2
%e 0 1 0 7 0 0 3 11 0 0 0 1 3 2 2 1 1
%e 0 1 0 8 0 0 0 16 0 0 0 1 0 7 0 3 0 2
%e 0 1 0 8 0 0 4 15 0 0 0 1 3 3 6 2 0 0 3
%e 0 1 0 9 0 0 0 21 0 0 0 2 0 9 0 7 0 1 0 4
%e 0 1 0 9 0 0 4 20 0 0 1 0 4 8 5 5 0 0 2 0 5
%e Row n = 14 counts the following partitions (A..E = 10..14):
%e (E) . (D1) . . (761) (B21) . . . . (6521) (8321) (7421)
%e (C2) (752) (A31) (6431)
%e (B3) (743) (941) (5432)
%e (A4) (932)
%e (95) (851)
%e (86) (842)
%e (653)
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,0,15},{k,0,n}]
%Y Row sums are A000009.
%Y Rightmost column n = k is A188431, non-strict A126796.
%Y The one-based weighted row sums are A284640.
%Y The corresponding rank statistic is A299701.
%Y The non-strict version is A365658.
%Y Central column n = 2k in the non-strict case is A365660.
%Y Reverse-weighted row-sums are A365922, non-strict A276024.
%Y A000041 counts integer partitions.
%Y A000124 counts distinct sums of subsets of {1..n}.
%Y A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y Cf. A046663, A108917, A122768, A137719, A304792, A364916.
%K nonn,tabl
%O 0,19
%A _Gus Wiseman_, Sep 28 2023