A353856 Triangle read by rows where T(n,k) is the number of integer compositions of n with run-sum trajectory (condensation) ending in a composition of length k.

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

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).

Links

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

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 length of the trajectory is A353854, firsts A072639, partitions A353841.

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.

A325268 counts partitions by omicron, rank statistic A304465.

A333489 ranks anti-runs, counted by A003242 (complement A261983).

A333627 ranks the run-lengths of standard compositions.

A353840-A353846 pertain to partition run-sum trajectory.

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, A325277, A333755, A353848, A353850, A353852.

Sequence in context: A238130 A238707 A181111 * A216800 A230831 A230899

Adjacent sequences: A353853 A353854 A353855 * A353857 A353858 A353859

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 01 2022

Status

approved