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

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: A029226 A093354 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