A353859 Triangle read by rows where T(n,k) is the number of integer compositions of n with composition run-sum trajectory of length k.

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

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).

Links

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

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

Column k = 1 is A003242, ranked by A333489, complement A261983.

Row sums are A011782.

Positive row-lengths are A070939.

The version for partitions is A353846, ranked by A353841.

This statistic (trajectory length) is ranked by A353854, firsts A072639.

Counting by length of last part instead of number of parts gives A353856.

A333627 ranks the run-lengths of standard compositions.

A353847 represents the run-sums of a composition, partitions A353832.

A353853-A353859 pertain to composition run-sum trajectory.

A353932 lists run-sums of standard compositions.

Cf. A237685, A238279, A304442, A304465, A318928, A325277, A333755, A353848, A353850, A353852, A353855, A353858.

Sequence in context: A144452 A217334 A369455 * A058865 A318502 A115090

Adjacent sequences: A353856 A353857 A353858 * A353860 A353861 A353862

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 02 2022

Status

approved