A332870 Number of compositions of n that are neither unimodal nor is their negation.

0, 0, 0, 0, 0, 0, 2, 9, 32, 92, 243, 587, 1361, 3027, 6564, 13928, 29127, 60180, 123300, 250945, 508326, 1025977, 2065437, 4150056, 8327344, 16692844, 33438984, 66951671, 134004892, 268148573, 536486146, 1073227893, 2146800237, 4294061970, 8588740071, 17178298617

Offset: 0

Author

Gus Wiseman, Mar 02 2020

Keywords

• nonn

Comments

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

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

Examples

			The a(6) = 2 and a(7) = 9 compositions:
  (1212)  (1213)
  (2121)  (1312)
          (2131)
          (3121)
          (11212)
          (12112)
          (12121)
          (21121)
          (21211)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

The case of run-lengths of partitions is A332640.

The version for unsorted prime signature is A332643.

Unimodal compositions are A001523.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Compositions whose negation is unimodal are A332578.

Compositions whose negation is not unimodal are A332669.

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

Compositions that are neither weakly increasing nor decreasing are A332834.

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

Cf. A000005, A000041, A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332641, A332746, A332831, A332833.

Programs

Mathematica

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!unimodQ[#]&&!unimodQ[-#]&]],{n,0,10}]

Formula

a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - Andrew Howroyd, Dec 30 2020

Extensions

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