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A072707
Number of non-unimodal compositions of n into distinct terms.
8
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
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.
Sequence in context: A227315 A080611 A171421 * A086105 A084701 A113815
KEYWORD
nonn,changed
AUTHOR
Henry Bottomley, Jul 04 2002
STATUS
approved