A332340 Number of widely alternately co-strongly normal compositions of n.

1, 1, 1, 3, 3, 4, 9, 11, 13, 23, 53, 78, 120, 207, 357, 707, 1183, 2030, 3558, 6229, 10868

Offset

0,4

Comments

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

Links

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

Example

The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (12)   (121)   (122)    (123)     (1213)     (1232)
             (21)   (211)   (212)    (132)     (1231)     (1322)
             (111)  (1111)  (1211)   (213)     (1312)     (2123)
                            (11111)  (231)     (1321)     (2132)
                                     (312)     (2122)     (2312)
                                     (321)     (2131)     (2321)
                                     (1212)    (2311)     (3122)
                                     (2121)    (3121)     (3212)
                                     (111111)  (3211)     (12131)
                                               (12121)    (13121)
                                               (1111111)  (21212)
                                                          (122111)
                                                          (11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).

Mathematica

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

Crossrefs

Normal compositions are A107429.

Compositions with normal run-lengths are A329766.

The Heinz numbers of the case of partitions are A332290.

The case of partitions is A332289.

The total (instead of alternating) version is A332337.

Not requiring normality gives A332338.

The strong version is this same sequence.

The narrow version is a(n) + 1 for n > 1.

Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.

Sequence in context: A060372 A128036 A332311 * A045794 A065678 A022598

Adjacent sequences: A332337 A332338 A332339 * A332341 A332342 A332343

Keyword

nonn,more

Author

Gus Wiseman, Feb 17 2020

Status

approved