A365661 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with a submultiset summing to k.

1, 1, 1, 1, 0, 1, 2, 1, 1, 2, 2, 1, 0, 1, 2, 3, 1, 1, 1, 1, 3, 4, 2, 2, 1, 2, 2, 4, 5, 2, 2, 2, 2, 2, 2, 5, 6, 3, 2, 3, 1, 3, 2, 3, 6, 8, 3, 3, 4, 3, 3, 4, 3, 3, 8, 10, 5, 4, 5, 4, 3, 4, 5, 4, 5, 10, 12, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 12

Offset

0,7

Comments

First differs from A284593 at T(6,3) = 1, A284593(6,3) = 2.

Rows are palindromic.

Are there only two zeros in the whole triangle?

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
  1
  1  1
  1  0  1
  2  1  1  2
  2  1  0  1  2
  3  1  1  1  1  3
  4  2  2  1  2  2  4
  5  2  2  2  2  2  2  5
  6  3  2  3  1  3  2  3  6
  8  3  3  4  3  3  4  3  3  8
Row n = 6 counts the following strict partitions:
  (6)      (5,1)    (4,2)    (3,2,1)  (4,2)    (5,1)    (6)
  (5,1)    (3,2,1)  (3,2,1)           (3,2,1)  (3,2,1)  (5,1)
  (4,2)                                                 (4,2)
  (3,2,1)                                               (3,2,1)
Row n = 10 counts the following strict partitions:
  A     91    82    73    64    532   64    73    82    91    A
  64    541   532   532   541   541   541   532   532   541   64
  73    631   721   631   631   4321  631   631   721   631   73
  82    721   4321  721   4321        4321  721   4321  721   82
  91    4321        4321                    4321        4321  91
  532                                                         532
  541                                                         541
  631                                                         631
  721                                                         721
  4321                                                        4321

Mathematica

Table[Length[Select[Select[IntegerPartitions[n], UnsameQ@@#&], MemberQ[Total/@Subsets[#], k]&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

Columns k = 0 and k = n are A000009.

The non-strict complement is A046663, central column A006827.

Central column n = 2k is A237258.

For subsets instead of partitions we have A365381.

The non-strict case is A365543.

The complement is A365663.

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

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

Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Sequence in context: A171533 A115236 A307777 * A284593 A190672 A327910

Adjacent sequences: A365658 A365659 A365660 * A365662 A365663 A365664

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 16 2023

Status

approved