A353836 Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums.

1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0, 0, 2, 5, 0, 0, 0, 0, 5, 5, 1, 0, 0, 0, 0, 2, 12, 1, 0, 0, 0, 0, 0, 7, 12, 3, 0, 0, 0, 0, 0, 0, 3, 19, 8, 0, 0, 0, 0, 0, 0, 0, 5, 27, 9, 1, 0, 0, 0, 0, 0, 0, 0, 2, 33, 20, 1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, May 26 2022

Keywords

• nonn
• tabl

Comments

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4).

Examples

			Triangle begins:
  1
  0  1
  0  2  0
  0  2  1  0
  0  4  1  0  0
  0  2  5  0  0  0
  0  5  5  1  0  0  0
  0  2 12  1  0  0  0  0
  0  7 12  3  0  0  0  0  0
  0  3 19  8  0  0  0  0  0  0
  0  5 27  9  1  0  0  0  0  0  0
  0  2 33 20  1  0  0  0  0  0  0  0
  0 13 28 34  2  0  0  0  0  0  0  0  0
  0  2 48 46  5  0  0  0  0  0  0  0  0  0
  0  5 65 51 14  0  0  0  0  0  0  0  0  0  0
  0  4 57 99 15  1  0  0  0  0  0  0  0  0  0  0
For example, row n = 8 counts the following partitions:
  (8)         (53)       (431)
  (44)        (62)       (521)
  (422)       (71)       (3221)
  (2222)      (332)
  (41111)     (611)
  (221111)    (3311)
  (11111111)  (4211)
              (5111)
              (22211)
              (32111)
              (311111)
              (2111111)
		

Crossrefs

Row sums are A000041.

Counting distinct parts instead of run-sums gives A116608.

Column k = 1 is A304442, ranked by A353833 (nonprime A353834).

The rank statistic is A353835, weak A353861, for compositions A353849.

A275870 counts collapsible partitions, ranked by A300273.

A351014 counts distinct runs in standard compositions.

A353832 represents the operation of taking run-sums of a partition.

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

A353840-A353846 pertain to partition run-sum trajectory.

A353864 counts rucksack partitions, ranked by A353866.

A353865 counts perfect rucksack partitions, ranked by A353867.

Cf. A008284, A047966, A071625, A165413, A175413, A325280, A333755.

Programs

Mathematica

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