A365660 -id:A365660 - cp's OEIS frontend

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4, 1, 1, 3, 0, 1, 5, 1, 0, 3, 0, 3, 0, 8, 1, 1, 3, 2, 2, 1, 2, 10, 1, 0, 5, 0, 3, 0, 5, 0, 16, 1, 1, 4, 0, 6, 2, 4, 2, 2, 20, 1, 0, 5, 0, 5, 0, 8, 0, 6, 0, 31, 1, 1, 6, 2, 3, 6, 6, 1, 4, 4, 4, 39, 1, 0, 6, 0, 6, 0, 12, 0, 8, 0, 13, 0, 55

Offset: 1

Gus Wiseman, Sep 16 2023

Conjecture: Positions of strictly positive rows are given by A048166.

			Triangle begins:
  1
  1  1
  1  0  2
  1  1  1  2
  1  0  2  0  4
  1  1  3  0  1  5
  1  0  3  0  3  0  8
  1  1  3  2  2  1  2 10
  1  0  5  0  3  0  5  0 16
  1  1  4  0  6  2  4  2  2 20
  1  0  5  0  5  0  8  0  6  0 31
  1  1  6  2  3  6  6  1  4  4  4 39
  1  0  6  0  6  0 12  0  8  0 13  0 55
  1  1  6  0  6  3 16  3  5  3  7  8  5 71

Robert Price, Table of n, a(n) for n = 1..300

Row sums are A000041.
Last column n = k is A126796.
Column k = 3 appears to be A137719.
This is the triangle for the rank statistic A299701.
Central column n = 2k is A365660.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A304792, A364272, A364916, A365381.

Table[Length[Select[IntegerPartitions[n],Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,10},{k,n}]

A365832 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k distinct sums of nonempty subsets.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 1, 1, 0, 0, 1, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 0, 4, 0, 0, 2, 2, 0, 0, 1, 0, 1, 0, 5, 0, 0, 0, 5, 0, 0, 0, 1, 0, 1, 0, 5, 0, 0, 2, 5, 0, 0, 0, 0, 2

Offset: 0

Gus Wiseman, Sep 28 2023

			The partition (7,6,1) has sums 1, 6, 7, 8, 13, 14, so is counted under T(14,6).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  0  1
  0  1  0  1  0
  0  1  0  2  0  0
  0  1  0  2  0  0  1
  0  1  0  3  0  0  0  1
  0  1  0  3  0  0  1  1  0
  0  1  0  4  0  0  0  3  0  0
  0  1  0  4  0  0  2  2  0  0  1
  0  1  0  5  0  0  0  5  0  0  0  1
  0  1  0  5  0  0  2  5  0  0  0  0  2
  0  1  0  6  0  0  0  8  0  0  0  1  0  2
  0  1  0  6  0  0  3  7  0  0  0  0  3  1  1
  0  1  0  7  0  0  0 12  0  0  0  1  0  4  0  2
  0  1  0  7  0  0  3 11  0  0  0  1  3  2  2  1  1
  0  1  0  8  0  0  0 16  0  0  0  1  0  7  0  3  0  2
  0  1  0  8  0  0  4 15  0  0  0  1  3  3  6  2  0  0  3
  0  1  0  9  0  0  0 21  0  0  0  2  0  9  0  7  0  1  0  4
  0  1  0  9  0  0  4 20  0  0  1  0  4  8  5  5  0  0  2  0  5
Row n = 14 counts the following partitions (A..E = 10..14):
  (E)  .  (D1)  .  .  (761)  (B21)  .  .  .  .  (6521)  (8321)  (7421)
          (C2)        (752)  (A31)              (6431)
          (B3)        (743)  (941)              (5432)
          (A4)               (932)
          (95)               (851)
          (86)               (842)
                             (653)

Row sums are A000009.
Rightmost column n = k is A188431, non-strict A126796.
The one-based weighted row sums are A284640.
The corresponding rank statistic is A299701.
The non-strict version is A365658.
Central column n = 2k in the non-strict case is A365660.
Reverse-weighted row-sums are A365922, non-strict A276024.
A000041 counts integer partitions.
A000124 counts distinct sums of subsets of {1..n}.
A365543 counts partitions with a submultiset summing to k, strict A365661.
Cf. A046663, A108917, A122768, A137719, A304792, A364916.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[Union[Total/@Rest[Subsets[#]]]]==k&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

A365658 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets.

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