A332337 Number of widely totally strongly normal compositions of n.

1, 1, 1, 3, 3, 3, 9, 9, 12, 23, 54, 77, 116, 205, 352, 697, 1174, 2013, 3538, 6209, 10830

Offset: 0

Author

Gus Wiseman, Feb 15 2020

Keywords

• nonn
• more

Comments

A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.

Examples

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (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). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).
		

Crossrefs

Normal compositions are A107429.

The case of partitions is A332278.

The non-strong version is A332279.

Heinz numbers in the case of partitions are A332291.

The narrow version is A332336.

The alternating version is A332340.

The co-strong version is this same sequence.

Cf. A025487, A100883, A181819, A317245, A317491, A329744, A332274, A332276, A332277, A332292, A332293, A332296.

Programs

Mathematica

totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

Formula

For n > 1, a(n) = A332336(n) - 1.