A332277 - OEIS

A332277

Number of widely totally normal integer partitions of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580

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OFFSET

0,4

COMMENTS

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

Also the number of widely totally normal reversed integer partitions of n.

LINKS

Table of n, a(n) for n=0..66.

EXAMPLE

The a(n) partitions for n = 1, 4, 10, 11, 16, 18:

1 211 4321 33221 443221 543321

1111 33211 322211 4432111 4333221

322111 332111 1111111111111111 4432221

1111111111 11111111111 4433211

43322211

44322111

111111111111111111

MATHEMATICA

recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];

Table[Length[Select[IntegerPartitions[n], recnQ]], {n, 0, 30}]

CROSSREFS

Normal partitions are A000009.

Taking multiplicities instead of run-lengths gives A317245.

Constantly recursively normal partitions are A332272.

The Heinz numbers of these partitions are A332276.

The case of all compositions (not just partitions) is A332279.

The co-strong version is A332278.

The recursive version is A332295.

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

Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.

Sequence in context: A093354 A374032 A317245 * A109701 A124751 A103374

Adjacent sequences: A332274 A332275 A332276 * A332278 A332279 A332280

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 12 2020

EXTENSIONS

a(61)-a(66) from Jinyuan Wang, Jun 26 2020

STATUS

approved