A365543 Triangle read by rows where T(n,k) is the number of integer partitions of n with a submultiset summing to k.

1, 1, 1, 2, 1, 2, 3, 2, 2, 3, 5, 3, 3, 3, 5, 7, 5, 5, 5, 5, 7, 11, 7, 8, 6, 8, 7, 11, 15, 11, 11, 11, 11, 11, 11, 15, 22, 15, 17, 15, 14, 15, 17, 15, 22, 30, 22, 23, 23, 22, 22, 23, 23, 22, 30, 42, 30, 33, 30, 33, 25, 33, 30, 33, 30, 42

Offset

0,4

Comments

Rows are palindromic.

Links

Table of n, a(n) for n=0..65.

Example

Triangle begins:
   1
   1   1
   2   1   2
   3   2   2   3
   5   3   3   3   5
   7   5   5   5   5   7
  11   7   8   6   8   7  11
  15  11  11  11  11  11  11  15
  22  15  17  15  14  15  17  15  22
  30  22  23  23  22  22  23  23  22  30
  42  30  33  30  33  25  33  30  33  30  42
  56  42  45  44  44  43  43  44  44  45  42  56
  77  56  62  58  62  56  53  56  62  58  62  56  77
Row n = 6 counts the following partitions:
  (6)       (51)      (42)      (33)      (42)      (51)      (6)
  (51)      (411)     (411)     (321)     (411)     (411)     (51)
  (42)      (321)     (321)     (3111)    (321)     (321)     (42)
  (411)     (3111)    (3111)    (2211)    (3111)    (3111)    (411)
  (33)      (2211)    (222)     (21111)   (222)     (2211)    (33)
  (321)     (21111)   (2211)    (111111)  (2211)    (21111)   (321)
  (3111)    (111111)  (21111)             (21111)   (111111)  (3111)
  (222)               (111111)            (111111)            (222)
  (2211)                                                      (2211)
  (21111)                                                     (21111)
  (111111)                                                    (111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 15}, {k, 0, n}]

Crossrefs

Columns k = 0 and k = n are A000041.

Central column n = 2k is A002219.

The complement is counted by A046663, strict A365663.

Row sums are A304792.

For subsets instead of partitions we have A365381.

The strict case is A365661.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A364272 counts sum-full strict partitions, sum-free A364349.

Cf. A088809, A093971, A122768, A108917, A299701, A364911, A365541, A365658.

Sequence in context: A342385 A367108 A283845 * A058071 A174961 A104889

Adjacent sequences: A365540 A365541 A365542 * A365544 A365545 A365546

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 16 2023

Status

approved