A353858 Number of integer compositions of n with run-sum trajectory ending in a singleton.

0, 1, 2, 2, 5, 2, 8, 2, 20, 5, 8, 2, 78, 2, 8, 8, 223, 2, 179, 2, 142, 8, 8, 2, 4808

Offset: 0

Author

Gus Wiseman, Jun 17 2022

Keywords

• nonn
• more

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 (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

Examples

			The a(0) = 0 through a(8) = 20 compositions:
  .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
          (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                       (112)            (222)                (224)
                       (211)            (1113)               (422)
                       (1111)           (2112)               (1124)
                                        (3111)               (2114)
                                        (11211)              (2222)
                                        (111111)             (4112)
                                                             (4211)
                                                             (11114)
                                                             (21122)
                                                             (22112)
                                                             (41111)
                                                             (111122)
                                                             (112112)
                                                             (211211)
                                                             (221111)
                                                             (1111211)
                                                             (1121111)
                                                             (11111111)
		

Crossrefs

The version for partitions is A353845, ranked by A353844.

The trajectory itself is A353853, last part A353855.

The lengths of trajectories of standard compositions are A353854.

This is column k = 1 of A353856, for partitions A353843.

These compositions are ranked by A353857.

A011782 counts compositions.

A066099 lists compositions in standard order.

A238279 and A333755 count compositions by number of runs.

A275870 counts collapsible partitions, ranked by A300273.

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

A353840-A353846 pertain to partition run-sum trajectory.

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

A353851 counts compositions with equal run-sums, ranked by A353848.

A353859 counts compositions by length of run-sum trajectory.

A353860 counts collapsible compositions.

A353932 lists run-sums of standard compositions.

Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]