A332336 Number of narrowly totally strongly normal compositions of n.

1, 1, 2, 4, 4, 4, 10, 10, 13, 24, 55, 78, 117, 206, 353, 698, 1175, 2014, 3539, 6210, 10831

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.

Links

Table of n, a(n) for n=0..20.

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

Normal compositions are A107429.

The non-strong version is A332296.

The case of partitions is A332297.

The co-strong version is A332336 (this sequence).

The wide version is A332337.

Cf. A025487, A316496, A317081, A317245, A317256, A317491, A329744, A332279, A332291, A332292, A332338, A332340.

Sequence in context: A082855 A260588 A107058 * A101449 A221564 A134188

Adjacent sequences: A332333 A332334 A332335 * A332337 A332338 A332339

Keyword

nonn,more

Author

Gus Wiseman, Feb 15 2020

Status

approved