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A332296 Number of narrowly totally normal compositions of n. 9
1, 1, 2, 4, 5, 7, 13, 23, 30, 63, 120, 209, 369, 651, 1198, 2174, 3896, 7023, 12699, 22941, 41565 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A sequence is narrowly totally normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) with narrowly totally normal run-lengths.

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) = A332279(n) + 1.

EXAMPLE

The a(0) = 1 through a(6) = 13 compositions:

  ()  (1)  (2)   (3)    (4)     (5)      (6)

           (11)  (12)   (112)   (122)    (123)

                 (21)   (121)   (212)    (132)

                 (111)  (211)   (221)    (213)

                        (1111)  (1121)   (231)

                                (1211)   (312)

                                (11111)  (321)

                                         (1212)

                                         (1221)

                                         (2112)

                                         (2121)

                                         (11211)

                                         (111111)

For example, starting with the composition (1,1,2,3,1,1) and repeatedly taking run-lengths gives (1,1,2,3,1,1) -> (2,1,1,2) -> (1,2,1) -> (1,1,1) -> (3). The first four are normal and the last is a singleton, so (1,1,2,3,1,1) is counted under a(9).

MATHEMATICA

tinQ[q_]:=Or[Length[q]<=1, And[Union[q]==Range[Max[q]], tinQ[Length/@Split[q]]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], tinQ]], {n, 0, 10}]

CROSSREFS

Normal compositions are A107429.

The wide version is A332279.

The wide recursive version (for partitions) is A332295.

The alternating version is A332296 (this sequence).

The strong version is A332336.

The co-strong version is (also) A332336.

Cf. A001462, A316496, A317081, A317245, A317491, A329744, A332276, A332277, A332278, A332297, A332337, A332340.

Sequence in context: A168540 A115008 A275368 * A115883 A063508 A101724

Adjacent sequences:  A332293 A332294 A332295 * A332297 A332298 A332299

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 15 2020

STATUS

approved

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Last modified March 30 21:56 EDT 2020. Contains 333132 sequences. (Running on oeis4.)