A332297 Number of narrowly totally strongly normal integer partitions of n.

1, 1, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2

Offset

0,3

Comments

A partition is narrowly totally strongly normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a narrowly totally strongly normal partition.

Links

Table of n, a(n) for n=0..80.

Example

The a(1) = 1, a(2) = 2, a(3) = 3, and a(55) = 4 partitions:
  (1)  (2)    (3)      (55)
       (1,1)  (2,1)    (10,9,8,7,6,5,4,3,2,1)
              (1,1,1)  (5,5,5,5,5,4,4,4,4,3,3,3,2,2,1)
                       (1)^55
For example, starting with the partition (3,3,2,2,1) and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2). The first four are normal and have weakly decreasing run-lengths, and the last is a singleton, so (3,3,2,2,1) is counted under a(11).

Mathematica

tinQ[q_]:=Or[q=={}, Length[q]==1, And[Union[q]==Range[Max[q]], GreaterEqual@@Length/@Split[q], tinQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n], tinQ]], {n, 0, 30}]

Crossrefs

Normal partitions are A000009.

The non-totally normal version is A316496.

The widely alternating version is A332292.

The non-strong case of compositions is A332296.

The case of compositions is A332336.

The wide version is a(n) - 1 for n > 1.

Cf. A001462, A025487, A100883, A181819, A182850, A317081, A317245, A317256, A317491, A332275, A332277, A332278, A332291, A332337.

Sequence in context: A218656 A200323 A075370 * A030350 A026240 A124474

Adjacent sequences: A332294 A332295 A332296 * A332298 A332299 A332300

Keyword

nonn,more

Author

Gus Wiseman, Feb 15 2020

Extensions

a(60)-a(80) from Jinyuan Wang, Jun 26 2020

Status

approved