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Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.
11

%I #13 Jun 05 2021 17:00:53

%S 1,0,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,1,2,1,1,0,1,1,1,2,1,1,0,1,1,2,3,2,

%T 1,1,0,1,1,2,1,3,2,1,1,0,1,1,2,2,4,3,2,1,1,0,1,1,2,3,1,4,3,2,1,1,0,1,

%U 1,3,3,4,6,4,3,2,1,1,0,1,1,1,1,3,1,6,4

%N Triangle read by rows where T(n,k) is the number of knapsack partitions of n with maximum k.

%C An integer partition is knapsack if every distinct submultiset has a different sum.

%H Fausto A. C. Cariboni, <a href="/A326017/b326017.txt">Table of n, a(n) for n = 0..10010</a>

%H Fausto A. C. Cariboni, <a href="/A326017/a326017.txt">Conjectures on columns of T(n,k)</a>, Jun 05 2021.

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 1 1 1

%e 0 1 1 1 1

%e 0 1 1 2 1 1

%e 0 1 1 1 2 1 1

%e 0 1 1 2 3 2 1 1

%e 0 1 1 2 1 3 2 1 1

%e 0 1 1 2 2 4 3 2 1 1

%e 0 1 1 2 3 1 4 3 2 1 1

%e 0 1 1 3 3 4 6 4 3 2 1 1

%e 0 1 1 1 1 3 1 6 4 3 2 1 1

%e 0 1 1 3 3 5 4 7 6 4 3 2 1 1

%e 0 1 1 2 3 5 4 1 7 6 4 3 2 1 1

%e 0 1 1 2 3 4 6 6 11 7 6 4 3 2 1 1

%e Row n = 9 counts the following partitions:

%e (111111111) (22221) (333) (432) (54) (63) (72) (81) (9)

%e (3222) (441) (522) (621) (711)

%e (531) (6111)

%e (51111)

%t ks[n_]:=Select[IntegerPartitions[n],UnsameQ@@Total/@Union[Subsets[#]]&];

%t Table[Length[Select[ks[n],Length[#]==k==0||Max@@#==k&]],{n,0,15},{k,0,n}]

%Y Row sums are A108917.

%Y Column k = 3 is A326034.

%Y Cf. A002033, A196723, A275972, A276024.

%Y Cf. A325592, A325857, A325862, A325863, A325864, A325877, A326016, A326018.

%K nonn,tabl

%O 0,19

%A _Gus Wiseman_, Jun 03 2019