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A332336 Number of narrowly totally strongly normal compositions of n. 9
1, 1, 2, 4, 4, 4, 10, 10, 13, 24, 55, 78, 117, 206, 353, 698, 1175, 2014, 3539, 6210, 10831 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 18 08:45 EDT 2021. Contains 348067 sequences. (Running on oeis4.)