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A332292 Number of widely alternately strongly normal integer partitions of n. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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.

LINKS

Table of n, a(n) for n=0..77.

EXAMPLE

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).

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}]

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.

Sequence in context: A097867 A075344 A144083 * A054350 A026606 A265918

Adjacent sequences:  A332289 A332290 A332291 * A332293 A332294 A332295

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 16 2020

EXTENSIONS

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

STATUS

approved

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Last modified May 8 06:11 EDT 2021. Contains 343653 sequences. (Running on oeis4.)