A353846 Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k.

1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0, 0, 3, 4, 0, 0, 0, 0, 4, 6, 1, 0, 0, 0, 0, 5, 9, 1, 0, 0, 0, 0, 0, 6, 11, 4, 1, 0, 0, 0, 0, 0, 8, 20, 2, 0, 0, 0, 0, 0, 0, 0, 10, 25, 7, 0, 0, 0, 0, 0, 0, 0, 0, 12, 37, 6, 1, 0, 0, 0, 0, 0, 0, 0

Offset

0,8

Comments

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking run-sums (or condensations) until a strict partition is reached. For example, the trajectory of (2,1,1) is (2,1,1) -> (2,2) -> (4).

Also the number of integer partitions of n with Kimberling's depth statistic (see A237685, A237750) equal to k-1.

Links

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

Example

Triangle begins:
   1
   0   1
   0   1   1
   0   2   1   0
   0   2   2   1   0
   0   3   4   0   0   0
   0   4   6   1   0   0   0
   0   5   9   1   0   0   0   0
   0   6  11   4   1   0   0   0   0
   0   8  20   2   0   0   0   0   0   0
   0  10  25   7   0   0   0   0   0   0   0
   0  12  37   6   1   0   0   0   0   0   0   0
   0  15  47  13   2   0   0   0   0   0   0   0   0
   0  18  67  15   1   0   0   0   0   0   0   0   0   0
   0  22  85  25   3   0   0   0   0   0   0   0   0   0   0
   0  27 122  26   1   0   0   0   0   0   0   0   0   0   0   0
For example, row n = 8 counts the following partitions (empty columns indicated by dots):
.  (8)    (44)        (422)     (4211)  .  .  .  .
   (53)   (332)       (32111)
   (62)   (611)       (41111)
   (71)   (2222)      (221111)
   (431)  (3221)
   (521)  (3311)
          (5111)
          (22211)
          (311111)
          (2111111)
          (11111111)

Mathematica

rsn[y_]:=If[y=={}, {}, NestWhileList[Reverse[Sort[Total/@ Split[Sort[#]]]]&, y, !UnsameQ@@#&]];
Table[Length[Select[IntegerPartitions[n], Length[rsn[#]]==k&]], {n, 0, 15}, {k, 0, n}]

Crossrefs

Row-sums are A000041.

Column k = 1 is A000009.

Column k = 2 is A237685.

Column k = 3 is A237750.

The version for run-lengths instead of run-sums is A225485 or A325280.

This statistic (trajectory length) is ranked by A353841 and A326371.

The version for compositions is A353859, see also A353847-A353858.

A005811 counts runs in binary expansion.

A275870 counts collapsible partitions, ranked by A300273.

A304442 counts partitions with all equal run-sums, ranked by A353833.

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

A353836 counts partitions by number of distinct run-sums.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353840-A353846 pertain to partition run-sum trajectory.

A353845 counts partitions whose run-sum trajectory ends in a singleton.

Cf. A008284, A047966, A181819, A325239, A325277, A353834, A353865.

Sequence in context: A122856 A328797 A328795 * A055791 A245842 A300574

Adjacent sequences: A353843 A353844 A353845 * A353847 A353848 A353849

Keyword

nonn,tabl

Author

Gus Wiseman, May 26 2022

Status

approved