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Number of non-unimodal compositions of n covering an initial interval of positive integers.
11

%I #6 Mar 02 2020 18:54:38

%S 0,0,0,0,0,1,5,14,35,83,193,417,890,1847,3809,7805,15833,32028,64513,

%T 129671,260155,521775,1044982,2092692,4188168,8381434,16767650,

%U 33544423,67098683,134213022,268443023,536912014,1073846768,2147720476,4295440133,8590833907

%N Number of non-unimodal compositions of n covering an initial interval of positive integers.

%C A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

%C A composition of n is a finite sequence of positive integers summing to n.

%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%F For n > 0, a(n) = A107429(n) - A227038(n).

%e The a(5) = 1 through a(7) = 14 compositions:

%e (212) (213) (1213)

%e (312) (1312)

%e (1212) (2113)

%e (2112) (2122)

%e (2121) (2131)

%e (2212)

%e (3112)

%e (3121)

%e (11212)

%e (12112)

%e (12121)

%e (21112)

%e (21121)

%e (21211)

%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&!unimodQ[#]&]],{n,0,10}]

%Y Not requiring non-unimodality gives A107429.

%Y Not requiring the covering condition gives A115981.

%Y The complement is counted by A227038.

%Y A version for partitions is A332579, with complement A332577.

%Y Unimodal compositions are A001523.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal normal sequences are A328509.

%Y Numbers whose unsorted prime signature is not unimodal are A332282.

%Y Cf. A007052, A072704, A072706, A332281, A332284, A332287, A332578, A332639, A332642, A332669, A332834, A332870.

%K nonn

%O 0,7

%A _Gus Wiseman_, Mar 02 2020