

A332289


Number of widely alternately costrongly normal integer partitions of n.


11



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

0,4


COMMENTS

An integer partition is widely alternately costrongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing runlengths (costrong) which, if reversed, are themselves a widely alternately costrongly normal partition.


LINKS

Table of n, a(n) for n=0..86.


EXAMPLE

The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:
(1) (21) (4321)
(111) (322111)
(1111111111)
For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking runlengths and reversing gives y > (3,2,1,1) > (2,1,1) > (2,1) > (1,1). These are all normal, have weakly increasing runlengths, and the last is all 1's, so y is counted a(14).


MATHEMATICA

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


CROSSREFS

Normal partitions are A000009.
Dominated by A317245.
The noncostrong version is A332277.
The total (instead of alternate) version is A332278.
The Heinz numbers of these partitions are A332290.
The strong version is A332292.
The case of reversed partitions is (also) A332292.
The generalization to compositions is A332340.
Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.
Sequence in context: A014710 A055174 A096369 * A102297 A269570 A243759
Adjacent sequences: A332286 A332287 A332288 * A332290 A332291 A332292


KEYWORD

nonn


AUTHOR

Gus Wiseman, Feb 13 2020


STATUS

approved



