A332278 Number of widely totally co-strongly normal integer partitions of n.

1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset

0,4

Comments

A sequence of integers is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal sequence.

Is this sequence bounded?

Links

Table of n, a(n) for n=0..78.

Example

The a(1) = 1 through a(20) = 2 partitions:
   1: (1)
   2: (11)
   3: (21),(111)
   4: (211),(1111)
   5: (11111)
   6: (321),(111111)
   7: (1111111)
   8: (11111111)
   9: (32211),(111111111)
  10: (4321),(322111),(1111111111)
  11: (11111111111)
  12: (111111111111)
  13: (1111111111111)
  14: (11111111111111)
  15: (54321),(111111111111111)
  16: (1111111111111111)
  17: (11111111111111111)
  18: (111111111111111111)
  19: (1111111111111111111)
  20: (4332221111),(11111111111111111111)

Mathematica

totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n], totnQ]], {n, 0, 30}]

Crossrefs

Not requiring co-strength gives A332277.

The strong version is A332297(n) - 1 for n > 1.

The narrow version is a(n) - 1 for n > 1.

The alternating version is A332289.

The Heinz numbers of these partitions are A332293.

The case of compositions is A332337.

Cf. A000009, A100883, A107429, A133808, A181819, A316496, A317245, A317491, A329746, A332279, A332290, A332291, A332292, A332296, A332576.

Sequence in context: A097456 A164002 A322156 * A182596 A087775 A366091

Adjacent sequences: A332275 A332276 A332277 * A332279 A332280 A332281

Keyword

nonn,more

Author

Gus Wiseman, Mar 05 2020

Extensions

a(71)-a(78) from Jinyuan Wang, Jun 26 2020

Status

approved