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

2,4

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.

Links

Table of n, a(n) for n=2..79.

Example

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)

Mathematica

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

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.

Sequence in context: A192295 A037195 A271319 * A319178 A376510 A254041

Adjacent sequences: A367401 A367402 A367403 * A367405 A367406 A367407

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 17 2023

Status

approved