OFFSET
0,3
COMMENTS
A sequence 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 sequence.
A composition of n is a finite sequence of positive integers summing to n.
FORMULA
For n > 1, a(n) = A332337(n) + 1.
EXAMPLE
The a(1) = 1 through a(8) = 13 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (112) (212) (123) (1213) (1232)
(21) (121) (221) (132) (1231) (2123)
(111) (1111) (11111) (213) (1312) (2132)
(231) (1321) (2312)
(312) (2131) (2321)
(321) (3121) (3212)
(1212) (11221) (12131)
(2121) (12121) (13121)
(111111) (1111111) (21212)
(22112)
(111221)
(11111111)
For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11) -> (2). The first four are normal with weakly decreasing run-lengths, and the last is a singleton, so (22112) is counted under a(8).
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[Join@@Permutations/@IntegerPartitions[n], tinQ]], {n, 0, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 15 2020
STATUS
approved