%I
%S 1,0,1,0,1,1,0,1,1,1,0,1,2,0,1,0,1,2,2,0,1,0,1,3,2,0,0,1,0,1,3,4,2,0,
%T 0,1,0,1,4,3,3,0,0,0,1,0,1,4,7,2,2,0,0,0,1,0,1,5,6,4,2,0,0,0,0,1,0,1,
%U 5,10,6,4,2,0,0,0,0,1,0,1,6,9,5,1,2,0,0,0,0,0,1
%N Triangle read by rows where T(n,k) is the number of lengthk knapsack partitions of n.
%C A knapsack partition of n is an integer partition of n whose distinct submultisets all have different sums.
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 1 1
%e 0 1 2 0 1
%e 0 1 2 2 0 1
%e 0 1 3 2 0 0 1
%e 0 1 3 4 2 0 0 1
%e 0 1 4 3 3 0 0 0 1
%e 0 1 4 7 2 2 0 0 0 1
%e 0 1 5 6 4 2 0 0 0 0 1
%e 0 1 5 10 6 4 2 0 0 0 0 1
%e 0 1 6 9 5 1 2 0 0 0 0 0 1
%e 0 1 6 14 10 5 2 2 0 0 0 0 0 1
%e 0 1 7 13 11 3 3 2 0 0 0 0 0 0 1
%e 0 1 7 19 16 7 3 2 2 0 0 0 0 0 0 1
%e Row n = 12 counts the following partitions (A = 10, B = 11, C = 12):
%e (C) (66) (444) (3333) (81111) (222222) (111111111111)
%e (75) (543) (5511) (711111)
%e (84) (552) (7221)
%e (93) (732) (7311)
%e (A2) (741) (9111)
%e (B1) (822)
%e (831)
%e (921)
%e (A11)
%t Table[Length[Select[IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[Subsets[#]]&]],{n,0,15},{k,0,n}]
%Y Row sums are A000041.
%Y Column k = 2 is A004526.
%Y Column k = 3 is A325690.
%Y Cf. A002219, A006827, A108917, A143823, A169942, A275972, A276024, A292886, A321143, A325676, A325687.
%K nonn,tabl
%O 0,13
%A _Gus Wiseman_, May 15 2019
