A353843 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory ending in a partition of length k. All zeros removed.

1, 1, 2, 2, 1, 4, 1, 2, 5, 5, 5, 1, 2, 12, 1, 8, 11, 3, 3, 19, 8, 5, 27, 9, 1, 2, 34, 19, 1, 15, 26, 34, 2, 2, 49, 45, 5, 5, 68, 48, 14, 4, 58, 98, 15, 1, 18, 76, 105, 31, 1, 2, 88, 159, 46, 2, 13, 98, 191, 79, 4, 2, 114, 261, 105, 8, 14, 148, 282, 164, 19

Offset: 0

Author

Gus Wiseman, Jun 04 2022

Keywords

• nonn
• tabf

Comments

The partition run-sum trajectory is obtained by repeatedly taking the run-sums until a strict partition is reached. For example, the trajectory of y = (3,2,1,1,1) is (3,2,1,1,1) -> (3,3,2) -> (6,2), so y is counted under T(8,2).

Examples

			Triangle begins:
   1
   1
   2
   2  1
   4  1
   2  5
   5  5  1
   2 12  1
   8 11  3
   3 19  8
   5 27  9  1
   2 34 19  1
  15 26 34  2
   2 49 45  5
   5 68 48 14
   4 58 98 15  1
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (4211)      (611)
  (41111)     (3311)
  (221111)    (5111)
  (11111111)  (22211)
              (32111)
              (311111)
              (2111111)
		

Crossrefs

Row sums are A000041.

Row-lengths are A003056.

The last part of the same trajectory is A353842.

Column k = 1 is A353845, compositions A353858.

The length of the trajectory is A353846.

The version for compositions is A353856.

A275870 counts collapsible partitions, ranked by A300273.

A304442 counts partitions with constant run-sums, ranked by A353833/A353834.

A325268 counts partitions by omicron, rank statistic A304465.

A353837 counts partitions with all distinct run-sums, ranked by A353838.

A353840-A353846 pertain to partition run-sum trajectory.

A353847 represents the run-sums of a composition, partitions A353832.

A353864 counts rucksack partitions, ranked by A353866.

A353865 counts perfect rucksack partitions, ranked by A353867.

A353932 lists run-sums of standard compositions.

Cf. A008284, A116608, A325242, A325268, A225485 or A325280.

Cf. A047966, A237685, A325277, A353841, A353853-A353859.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], Length[FixedPoint[Sort[Total/@Split[#]]&,#]]==k&]],{n,0,15},{k,0,n}]