A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

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

Offset: 0

Author

Gus Wiseman, Feb 29 2020

Keywords

• nonn

Comments

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

Examples

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Eric Weisstein's World of Mathematics, Unimodal Sequence.

Crossrefs

The case of partitions is A332641.

The version for unsorted prime signature is A332831.

The version for the compositions themselves (not run-lengths) is A332834.

The complement is counted by A332835.

Unimodal compositions are A001523.

Partitions with weakly increasing run-lengths are A100883.

Compositions that are not unimodal are A115981.

Compositions with equal run-lengths are A329738.

Compositions whose run-lengths are unimodal are A332726.

Compositions whose run-lengths are not unimodal are A332727.

Partitions with weakly increasing or weakly decreasing run-lengths: A332745.

Compositions with weakly increasing run-lengths 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.

Programs

Mathematica

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

Formula

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

Extensions

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