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Triangle read by rows where T(n,k) is the number of strict integer partitions of n with two distinct parts summing to k.
9

%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