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.

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.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 15 2020

EXTENSIONS

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

STATUS

approved