A332279 - OEIS

A332279

Number of widely totally normal compositions of n.

1, 1, 1, 3, 4, 6, 12, 22, 29, 62, 119, 208, 368, 650, 1197, 2173, 3895, 7022, 12698, 22940, 41564

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OFFSET

0,4

COMMENTS

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

A composition of n is a finite sequence of positive integers with sum n.

LINKS

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

FORMULA

For n > 1, a(n) = A332296(n) - 1.

EXAMPLE

The a(1) = 1 through a(7) = 22 compositions:

(1) (11) (12) (112) (122) (123) (1123)

(21) (121) (212) (132) (1132)

(111) (211) (221) (213) (1213)

(1111) (1121) (231) (1231)

(1211) (312) (1312)

(11111) (321) (1321)

(1212) (2113)

(1221) (2122)

(2112) (2131)

(2121) (2212)

(11211) (2311)

(111111) (3112)

(3121)

(3211)

(11221)

(12112)

(12121)

(12211)

(21121)

(111211)

(112111)

(1111111)

For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).

MATHEMATICA

recnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], recnQ[Length/@Split[ptn]]]];

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

CROSSREFS

Normal compositions are A107429.

Constantly recursively normal partitions are A332272.

The case of partitions is A332277.

The case of reversed partitions is (also) A332277.

The narrow version is A332296.

The strong version is A332337.

The co-strong version is (also) A332337.