A354909 Number of integer compositions of n that are not the run-sums of any other composition.

0, 0, 1, 1, 3, 7, 16, 33, 74, 155, 329, 688, 1439, 2975, 6154, 12654, 25964, 53091, 108369, 220643, 448520

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).

Links

Table of n, a(n) for n=0..20.

Example

The a(0) = 0 through a(6) = 16 compositions:
  .  .  (11)  (111)  (112)   (113)    (114)
                     (211)   (311)    (411)
                     (1111)  (1112)   (1113)
                             (1121)   (1122)
                             (1211)   (1131)
                             (2111)   (1221)
                             (11111)  (1311)
                                      (2112)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

Mathematica

Table[Length[Complement[Join@@Permutations/@IntegerPartitions[n], Total/@Split[#]&/@Join@@Permutations/@IntegerPartitions[n]]], {n, 0, 15}]

Crossrefs

The version for binary words is A000918, complement A000126.

These compositions are ranked by A354904 = positions of zeros in A354578.

The complement is counted by A354910, ranked by A354912.

A003242 counts anti-run compositions, ranked by A333489.

A238279 and A333755 count compositions by number of runs.

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

A353853-A353859 pertain to composition run-sum trajectory.

A353932 lists run-sums of standard compositions, rows ranked by A353847.

Cf. A005811, A027336, A066099, A124767, A274174, A351597, A353849, A353850, A353860, A354905, A354907.

Sequence in context: A217942 A002936 A014668 * A182615 A181893 A054455

Adjacent sequences: A354906 A354907 A354908 * A354910 A354911 A354912

Keyword

nonn,more

Author

Gus Wiseman, Jun 19 2022

Status

approved