A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3

Offset: 0

Author

Gus Wiseman, Nov 07 2023

Keywords

• nonn
• tabf

Examples

			The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
    0   1
    1   1
    2   2   1
    5   3   3
    8   4   9   1
   17   6  16   1   2
   24   7  33   4   9
   46  11  52   3  18   1   4
   64  12  91   6  38   3  15   1   1
  107  17 138   9  68   2  28   2  12   0   2
  147  19 219  12 117   6  56   3  34   2   9   0   3
Row n = 4 counts the following partitions:
  (8)     (44)        (431)      (221111)
  (71)    (3311)      (422)
  (62)    (2222)      (4211)
  (611)   (11111111)  (41111)
  (53)                (3221)
  (521)               (32111)
  (5111)              (311111)
  (332)               (22211)
                      (2111111)
		

Crossrefs

Row sums w/o the first column are A002219, ranks A357976, strict A237258.

Column k = 0 is A006827.

Row sums are A058696.

Column k = 1 is A108917.

The corresponding rank statistic is A357879 (without empty rows).

A000041 counts integer partitions, strict A000009.

A182616 counts partitions of 2n that do not contain n, ranks A366321.

A182616 counts partitions of 2n with at least one odd part, ranks A366530.

A276024 counts positive subset-sums of partitions, strict A284640.

A304792 counts subset-sums of partitions, rank statistic A299701.

A365543 counts partitions of n with a submultiset summing to k.

Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

Programs

Mathematica

t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]

Formula

T(n,1) = A108917(n).