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

Offset

3,6

Links

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

Example

Triangle begins:
  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
  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
                                                            76

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