A367405 - OEIS

A367405

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

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, 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, 6, 5, 6, 5, 7, 5, 7, 4, 5, 6, 5, 5, 7, 7, 8, 7, 8, 8, 7, 7, 5, 5, 7

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OFFSET

3,6

LINKS

Table of n, a(n) for n=3..93.

EXAMPLE

Triangle begins:

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 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 6 5 6 5 7 5 7 4 5 6

5 5 7 7 8 7 8 8 7 7 5 5 7

6 5 9 8 10 7 10 9 10 7 9 5 6 7

7 7 10 10 12 11 11 11 12 10 9 9 6 6 8

9 7 13 11 15 12 13 13 15 13 13 9 11 6 7 8

Row n = 9 counts the following strict partitions:

(6,2,1) (5,3,1) (4,3,2) (5,3,1) (6,2,1) (6,2,1) (8,1)

(4,3,2) (4,3,2) (5,3,1) (7,2)

(6,3)

(5,4)

Row n = 13 counts the following strict partitions (A=10, B=11, C=12):

A21 931 841 751 652 751 841 931 A21 A21 C1

7321 7321 832 742 643 7321 742 832 832 931 B2

6421 5431 7321 6421 6421 652 7321 7321 742 841 A3

6421 5431 5431 6421 643 643 652 751 94

5431 5431 5431 6421 85

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#, {2}], k]&]], {n, 3, 10}, {k, 3, n}]

CROSSREFS

Column n = k is A004526.

Column k = 3 is A025148.

For subsets instead of partitions we have A365541, non-binary A365381.

The non-binary version is A365661, non-strict A365543.

The non-binary complement is A365663, non-strict A046663.

Row sums are A366741, non-strict A366738.

The non-strict version is A367404.

Cf. A000041, A088809, A093971, A122768, A108917, A284640, A304792, A364272, A364911, A365658.

Sequence in context: A362451 A092673 A243842 * A112400 A316523 A219185

Adjacent sequences: A367402 A367403 A367404 * A367406 A367407 A367408

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Nov 18 2023

STATUS

approved