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

Author

Gus Wiseman, Jun 19 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).

Examples

			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)
		

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.

Programs

Mathematica

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