A332292 Number of widely alternately strongly normal integer partitions of n.

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

Offset: 0

Author

Gus Wiseman, Feb 16 2020

Keywords

• nonn
• more

Comments

An integer partition is widely alternately strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which, if reversed, are themselves a widely alternately strongly normal partition.

Also the number of widely alternately co-strongly normal reversed integer partitions of n.

Examples

			The a(1) = 1, a(3) = 2, and a(21) = 3 partitions:
  (1)  (21)   (654321)
       (111)  (4443321)
              (111111111111111111111)
For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly decreasing run-lengths, and the last is all 1's, so y is counted under a(21).
		

Crossrefs

Normal partitions are A000009.

The non-strong version is A332277.

The co-strong version is A332289.

The case of reversed partitions is (also) A332289.

The case of compositions is A332340.

Cf. A100883, A181819, A317081, A317245, A317256, A317491, A329746, A329747, A332278, A332290, A332291, A332297, A332337.

Programs

Mathematica

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

Extensions

a(71)-a(77) from Jinyuan Wang, Jun 26 2020