

A332833


Number of compositions of n whose runlengths are neither weakly increasing nor weakly decreasing.


20



0, 0, 0, 0, 0, 0, 3, 8, 27, 75, 185, 441, 1025, 2276, 4985, 10753, 22863, 48142, 100583, 208663, 430563, 884407, 1809546, 3690632, 7506774, 15233198, 30851271, 62377004, 125934437, 253936064, 511491634, 1029318958, 2069728850, 4158873540, 8351730223, 16762945432
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OFFSET

0,7


COMMENTS

A composition of n is a finite sequence of positive integers summing to n.


LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000
MathWorld, Unimodal Sequence


FORMULA

a(n) = 2^(n  1)  2 * A332836(n) + A329738(n).


EXAMPLE

The a(6) = 3 and a(7) = 8 compositions:
(1221) (2113)
(2112) (3112)
(11211) (11311)
(12112)
(21112)
(21121)
(111211)
(112111)


MATHEMATICA

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !Or[LessEqual@@Length/@Split[#], GreaterEqual@@Length/@Split[#]]&]], {n, 0, 10}]


CROSSREFS

The case of partitions is A332641.
The version for unsorted prime signature is A332831.
The version for the compositions themselves (not runlengths) is A332834.
The complement is counted by A332835.
Unimodal compositions are A001523.
Partitions with weakly increasing runlengths are A100883.
Compositions that are not unimodal are A115981.
Compositions with equal runlengths are A329738.
Compositions whose runlengths are unimodal are A332726.
Compositions whose runlengths are not unimodal are A332727.
Partitions with weakly increasing or weakly decreasing runlengths: A332745.
Compositions with weakly increasing runlengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.
Sequence in context: A066023 A347830 A305049 * A148823 A205503 A242537
Adjacent sequences: A332830 A332831 A332832 * A332834 A332835 A332836


KEYWORD

nonn


AUTHOR

Gus Wiseman, Feb 29 2020


EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020


STATUS

approved



