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A332870
Number of compositions of n that are neither unimodal nor is their negation.
17
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
0,7
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
FORMULA
a(n) = 2^(n-1) - A001523(n) - A332578(n) + 2*A000041(n) - A000005(n) for n > 0. - Andrew Howroyd, Dec 30 2020
EXAMPLE
The a(6) = 2 and a(7) = 9 compositions:
(1212) (1213)
(2121) (1312)
(2131)
(3121)
(11212)
(12112)
(12121)
(21121)
(21211)
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}]
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.
Sequence in context: A242512 A198016 A082114 * A074084 A053152 A077644
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 02 2020
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved