A332834 Number of compositions of n that are neither weakly increasing nor weakly decreasing.

0, 0, 0, 0, 1, 4, 14, 36, 88, 199, 432, 914, 1900, 3896, 7926, 16036, 32311, 64944, 130308, 261166, 523040, 1046996, 2095152, 4191796, 8385466, 16773303, 33549564, 67102848, 134210298, 268426328, 536859712, 1073728142, 2147466956, 4294947014, 8589909976

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(4) = 1 through a(6) = 14 compositions:
  (121)  (131)   (132)
         (212)   (141)
         (1121)  (213)
         (1211)  (231)
                 (312)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11121)
                 (11211)
                 (12111)
		

Links

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

Crossrefs

The version for unsorted prime signature is A332831.

The version for run-lengths of compositions is A332833.

The complement appears to be counted by A329398.

Unimodal compositions are A001523.

Compositions that are not unimodal are A115981.

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

Compositions with weakly increasing or decreasing run-lengths are A332835.

Compositions with weakly increasing run-lengths are A332836.

Compositions that are neither unimodal nor is their negation are A332870.

Cf. A007052, A072704, A107429, A328509, A329744, A332281, A332284, A332578, A332640, A332641, A332643, A332669, A332746.

Programs

Mathematica

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

PARI

a(n)={if(n==0, 0, 2^(n-1) - 2*numbpart(n) + numdiv(n))} \\ Andrew Howroyd, Dec 30 2020

Formula

a(n) = 2^(n - 1) - 2 * A000041(n) + A000005(n).