

A332669


Number of compositions of n whose negation is not unimodal.


37



0, 0, 0, 0, 1, 3, 11, 28, 71, 165, 372, 807, 1725, 3611, 7481, 15345, 31274, 63392, 128040, 257865, 518318, 1040277, 2085714, 4178596, 8367205, 16748151, 33515214, 67056139, 134147231, 268341515, 536746350, 1073577185, 2147266984, 4294683056, 8589563136, 17179385180
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OFFSET

0,6


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.


LINKS

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


FORMULA

a(n) + A332578(n) = 2^(n  1) for n > 0.


EXAMPLE

The a(4) = 1 through a(6) = 11 compositions:
(121) (131) (132)
(1121) (141)
(1211) (231)
(1131)
(1212)
(1221)
(1311)
(2121)
(11121)
(11211)
(12111)


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[#]&]], {n, 0, 10}]


CROSSREFS

The strict case is A072707.
The complement is counted by A332578.
The version for runlengths of partitions is A332639.
The version for unsorted prime signature is A332642.
The version for 0appended firstdifferences of partitions is A332744.
The case that is not unimodal either is A332870.
Unimodal compositions are A001523.
Nonunimodal permutations are A059204.
Nonunimodal compositions are A115981.
Nonunimodal normal sequences are A328509.
Numbers whose unsorted prime signature is not unimodal are A332282.
A triangle for compositions with unimodal negation is A332670.
Cf. A007052, A072706, A227038, A329398, A332281, A332284, A332638, A332728, A332742, A332832.
Sequence in context: A182260 A163696 A092781 * A302509 A335899 A018743
Adjacent sequences: A332666 A332667 A332668 * A332670 A332671 A332672


KEYWORD

nonn


AUTHOR

Gus Wiseman, Feb 28 2020


EXTENSIONS

Terms a(21) and beyond from Andrew Howroyd, Mar 01 2020


STATUS

approved



