OFFSET
0,5
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 the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,1,1,3,1,1,2,1,1,2,1) -> (2,2,3,2,2,2,2,1) -> (4,3,8,1) is counted under T(15,4).
EXAMPLE
Triangle begins:
1
0 1
0 2 0
0 2 2 0
0 5 2 1 0
0 2 12 2 0 0
0 8 10 12 2 0 0
0 2 32 23 6 1 0 0
0 20 26 51 28 3 0 0 0
0 5 66 109 52 22 2 0 0 0
0 8 108 144 188 53 10 1 0 0 0
0 2 134 358 282 196 48 4 0 0 0 0
For example, row n = 6 counts the following compositions:
. (6) (15) (123) (1212) . .
(33) (24) (132) (2121)
(222) (42) (141)
(1113) (51) (213)
(2112) (114) (231)
(3111) (411) (312)
(11211) (1122) (321)
(111111) (2211) (1131)
(11112) (1221)
(21111) (1311)
(11121)
(12111)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&, #]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Row sums are A011782.
Row-lengths without zeros appear to be A131737.
The version for partitions is A353843.
The last part of the same trajectory is A353855.
Column k = 1 is A353858.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A333627 ranks the run-lengths of standard compositions.
A353932 lists run-sums of standard compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 01 2022
STATUS
approved