%I #6 Nov 19 2023 10:34:28
%S 1,0,1,0,0,2,1,1,1,2,1,0,1,1,3,1,1,1,1,2,3,1,1,1,2,2,2,4,2,2,3,2,3,2,
%T 3,4,2,2,3,2,3,3,3,3,5,3,2,4,3,4,4,5,3,4,5,3,3,5,4,4,5,5,5,4,4,6,4,3,
%U 6,5,6,5,7,5,7,4,5,6,5,5,7,7,8,7,8,8,7,7,5,5,7
%N Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.
%e Triangle begins:
%e 1
%e 0 1
%e 0 0 2
%e 1 1 1 2
%e 1 0 1 1 3
%e 1 1 1 1 2 3
%e 1 1 1 2 2 2 4
%e 2 2 3 2 3 2 3 4
%e 2 2 3 2 3 3 3 3 5
%e 3 2 4 3 4 4 5 3 4 5
%e 3 3 5 4 4 5 5 5 4 4 6
%e 4 3 6 5 6 5 7 5 7 4 5 6
%e 5 5 7 7 8 7 8 8 7 7 5 5 7
%e 6 5 9 8 10 7 10 9 10 7 9 5 6 7
%e 7 7 10 10 12 11 11 11 12 10 9 9 6 6 8
%e 9 7 13 11 15 12 13 13 15 13 13 9 11 6 7 8
%e Row n = 9 counts the following strict partitions:
%e (6,2,1) (5,3,1) (4,3,2) (5,3,1) (6,2,1) (6,2,1) (8,1)
%e (4,3,2) (4,3,2) (5,3,1) (7,2)
%e (6,3)
%e (5,4)
%e Row n = 13 counts the following strict partitions (A=10, B=11, C=12):
%e A21 931 841 751 652 751 841 931 A21 A21 C1
%e 7321 7321 832 742 643 7321 742 832 832 931 B2
%e 6421 5431 7321 6421 6421 652 7321 7321 742 841 A3
%e 6421 5431 5431 6421 643 643 652 751 94
%e 5431 5431 5431 6421 85
%e 76
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], k]&]], {n,3,10}, {k,3,n}]
%Y Column n = k is A004526.
%Y Column k = 3 is A025148.
%Y For subsets instead of partitions we have A365541, non-binary A365381.
%Y The non-binary version is A365661, non-strict A365543.
%Y The non-binary complement is A365663, non-strict A046663.
%Y Row sums are A366741, non-strict A366738.
%Y The non-strict version is A367404.
%Y Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.
%K nonn,tabl
%O 3,6
%A _Gus Wiseman_, Nov 18 2023