A332874 Number of strict compositions of n that are neither unimodal nor is their negation.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 10, 20, 30, 50, 150, 180, 290, 420, 630, 860, 1828, 2168, 3326, 4514, 6530, 8576, 12188, 20096, 25314, 35576, 48062, 65592, 86752, 117222, 152060, 237590, 292346, 402798, 524596, 711270, 910606, 1221204, 1554382, 2044460, 2927124

Offset

0,11

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. It is strict if there are not repeated parts.

Links

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

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

G.f.: Sum_{k>=4} (k! - 2^k + 2) * [y^k](Product_{j>=1} 1 + y*x^j). - Andrew Howroyd, Apr 16 2021

Example

The a(10) = 10 through a(12) = 20 compositions:
  (1,3,2,4)  (1,3,2,5)  (1,3,2,6)
  (1,4,2,3)  (1,5,2,3)  (1,4,2,5)
  (2,1,4,3)  (2,1,5,3)  (1,5,2,4)
  (2,3,1,4)  (2,3,1,5)  (1,6,2,3)
  (2,4,1,3)  (2,5,1,3)  (2,1,5,4)
  (3,1,4,2)  (3,1,5,2)  (2,1,6,3)
  (3,2,4,1)  (3,2,5,1)  (2,3,1,6)
  (3,4,1,2)  (3,5,1,2)  (2,4,1,5)
  (4,1,3,2)  (5,1,3,2)  (2,5,1,4)
  (4,2,3,1)  (5,2,3,1)  (2,6,1,3)
                        (3,1,6,2)
                        (3,2,6,1)
                        (3,6,1,2)
                        (4,1,5,2)
                        (4,2,5,1)
                        (4,5,1,2)
                        (5,1,4,2)
                        (5,2,4,1)
                        (6,1,3,2)
                        (6,2,3,1)

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], UnsameQ@@#&&!unimodQ[#]&&!unimodQ[-#]&]], {n, 0, 20}]

Prog

(PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=4, n, (k! - 2^k + 2)*polcoef(p, k, y)), -(n+1))} \\ Andrew Howroyd, Apr 16 2021

Crossrefs

The non-strict version for unsorted prime signature is A332643.

The non-strict version is A332870.

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.

Compositions with neither weakly increasing nor weakly decreasing run-lengths are A332833.

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

Cf. A007052, A072704, A227038, A329398, A332281, A332284, A332639, A332640, A332641, A332745, A332746, A332831, A332834.

Sequence in context: A309464 A368362 A022093 * A076817 A324494 A344104

Adjacent sequences: A332871 A332872 A332873 * A332875 A332876 A332877

Keyword

nonn

Author

Gus Wiseman, Mar 04 2020

Extensions

Terms a(21) and beyond from Andrew Howroyd, Apr 16 2021

Status

approved