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).
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.
A005811 counts runs in binary expansion.
A353832 represents the operation of taking run-sums of a partition
A353836 counts partitions by number of distinct run-sums.
A353845 counts partitions whose run-sum trajectory ends in a singleton.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 26 2022
STATUS
approved