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 the run-sums transformation (or condensation, represented by A353847) until an anti-run is reached. For example, the trajectory (2,4,2,1,1) -> (2,4,2,2) -> (2,4,4) -> (2,8) is counted under T(10,4).
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 3 1 0
0 4 2 2 0
0 7 7 2 0 0
0 14 14 4 0 0 0
0 23 29 12 0 0 0 0
0 39 56 25 8 0 0 0 0
0 71 122 53 10 0 0 0 0 0
0 124 246 126 16 0 0 0 0 0 0
0 214 498 264 48 0 0 0 0 0 0 0
For example, row n = 5 counts the following compositions:
(5) (113) (1121)
(14) (122) (1211)
(23) (221)
(32) (311)
(41) (1112)
(131) (2111)
(212) (11111)
MATHEMATICA
rsc[y_]:=If[y=={}, {}, NestWhileList[Total/@Split[#]&, y, MatchQ[#, {___, x_, x_, ___}]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Length[rsc[#]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Row sums are A011782.
Positive row-lengths are A070939.
Counting by length of last part instead of number of parts gives A353856.
A333627 ranks the run-lengths of standard compositions.
A353932 lists run-sums of standard compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 02 2022
STATUS
approved