A072707 Number of non-unimodal compositions of n into distinct terms.

0, 0, 0, 0, 0, 0, 2, 2, 4, 6, 24, 26, 46, 64, 100, 224, 276, 416, 590, 850, 1144, 2214, 2644, 3938, 5282, 7504, 9776, 13704, 21984, 27632, 38426, 51562, 69844, 91950, 123504, 159658, 246830, 303400, 416068, 540480, 730268, 933176, 1248110

Offset

0,7

Comments

Also the number of compositions of n into distinct terms whose negation is not unimodal. - Gus Wiseman, Mar 05 2020

Links

Table of n, a(n) for n=0..42.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) = A032020(n) - A072706(n) = Sum_{k} A059204(k) * A060016(n, k).

Example

a(6)=2 since 6 can be written as 2+1+3 or 3+1+2.
From Gus Wiseman, Mar 05 2020: (Start)
The a(6) = 2 through a(9) = 6 strict compositions:
  (2,1,3)  (2,1,4)  (2,1,5)  (2,1,6)
  (3,1,2)  (4,1,2)  (3,1,4)  (3,1,5)
                    (4,1,3)  (3,2,4)
                    (5,1,2)  (4,2,3)
                             (5,1,3)
                             (6,1,2)
(End)

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[#]&]], {n, 0, 16}] (* Gus Wiseman, Mar 05 2020 *)

Crossrefs

The complement is counted by A072706.

The non-strict version is A115981.

The case where the negation is not unimodal either is A332874.

Unimodal compositions are A001523.

Strict compositions are A032020.

Non-unimodal permutations are A059204.

A triangle for strict unimodal compositions is A072705.

Non-unimodal sequences covering an initial interval are A328509.

Numbers whose prime signature is not unimodal are A332282.

Strict partitions whose 0-appended differences are not unimodal are A332286.

Compositions whose negation is unimodal are A332578.

Compositions whose negation is not unimodal are A332669.

Non-unimodal compositions covering an initial interval are A332743.

Cf. A007052, A072704, A107429, A227038, A329398, A332281, A332284, A332285, A332579, A332870.

Sequence in context: A227315 A080611 A171421 * A086105 A084701 A113815

Adjacent sequences: A072704 A072705 A072706 * A072708 A072709 A072710

Keyword

nonn

Author

Henry Bottomley, Jul 04 2002

Status

approved