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A332297
Number of narrowly totally strongly normal integer partitions of n.
12
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.
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.
Sequence in context: A218656 A200323 A075370 * A030350 A026240 A124474
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 15 2020
EXTENSIONS
a(60)-a(80) from Jinyuan Wang, Jun 26 2020
STATUS
approved