A367404 Triangle read by rows where T(n,k) is the number of integer partitions of n with a semi-sum k.

1, 1, 1, 2, 1, 2, 3, 2, 2, 2, 5, 3, 4, 2, 3, 7, 5, 6, 4, 3, 3, 11, 7, 9, 6, 6, 3, 4, 15, 11, 13, 10, 9, 6, 4, 4, 22, 15, 20, 13, 15, 9, 8, 4, 5, 30, 22, 27, 21, 21, 15, 12, 8, 5, 5, 42, 30, 39, 28, 30, 21, 20, 12, 10, 5, 6, 56, 42, 53, 41, 42, 33, 28, 20, 15, 10, 6, 6

Offset: 2

Author

Gus Wiseman, Nov 17 2023

Keywords

• nonn
• tabl

Comments

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

Examples

			The partition y = (3,2,1,1) has semi-sum 3 = 2+1, but no semi-sum 6, so y is counted under T(7,3) but not under T(7,6).
Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   2
   5   3   4   2   3
   7   5   6   4   3   3
  11   7   9   6   6   3   4
  15  11  13  10   9   6   4   4
  22  15  20  13  15   9   8   4   5
  30  22  27  21  21  15  12   8   5   5
  42  30  39  28  30  21  20  12  10   5   6
  56  42  53  41  42  33  28  20  15  10   6   6
  77  56  73  55  60  42  44  28  25  15  12   6   7
Row n = 7 counts the following partitions:
  (511)      (421)     (331)    (421)   (511)  (61)
  (4111)     (3211)    (322)    (4111)  (421)  (52)
  (3211)     (2221)    (3211)   (322)   (331)  (43)
  (31111)    (22111)   (31111)  (3211)
  (22111)    (211111)  (2221)
  (211111)             (22111)
  (1111111)
		

Crossrefs

Column k = 0 is A000041.

Column n = k is A004526.

The complement for all submultisets is A046663, strict A365663.

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

The non-binary version is A365543, strict A365661.

Row sums are A366738.

The strict case is A367405.

Cf. A122768, A108917, A299701, A304792, A364272, A364911, A365658.

Programs

Mathematica

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