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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.

%I A365658 #13 Apr 08 2025 13:21:27
%S A365658 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,
%T A365658 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,
%U A365658 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
%N 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.
%C A365658 Conjecture: Positions of strictly positive rows are given by A048166.
%H A365658 Robert Price, <a href="/A365658/b365658.txt">Table of n, a(n) for n = 1..300</a>
%e A365658 Triangle begins:
%e A365658   1
%e A365658   1  1
%e A365658   1  0  2
%e A365658   1  1  1  2
%e A365658   1  0  2  0  4
%e A365658   1  1  3  0  1  5
%e A365658   1  0  3  0  3  0  8
%e A365658   1  1  3  2  2  1  2 10
%e A365658   1  0  5  0  3  0  5  0 16
%e A365658   1  1  4  0  6  2  4  2  2 20
%e A365658   1  0  5  0  5  0  8  0  6  0 31
%e A365658   1  1  6  2  3  6  6  1  4  4  4 39
%e A365658   1  0  6  0  6  0 12  0  8  0 13  0 55
%e A365658   1  1  6  0  6  3 16  3  5  3  7  8  5 71
%t A365658 Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]
%Y A365658 Row sums are A000041.
%Y A365658 Last column n = k is A126796.
%Y A365658 Column k = 3 appears to be A137719.
%Y A365658 This is the triangle for the rank statistic A299701.
%Y A365658 Central column n = 2k is A365660.
%Y A365658 A000009 counts subsets summing to n.
%Y A365658 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365658 A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y A365658 Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.
%K A365658 nonn,tabl
%O A365658 1,6
%A A365658 _Gus Wiseman_, Sep 16 2023