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