A332743 Number of non-unimodal compositions of n covering an initial interval of positive integers.

0, 0, 0, 0, 0, 1, 5, 14, 35, 83, 193, 417, 890, 1847, 3809, 7805, 15833, 32028, 64513, 129671, 260155, 521775, 1044982, 2092692, 4188168, 8381434, 16767650, 33544423, 67098683, 134213022, 268443023, 536912014, 1073846768, 2147720476, 4295440133, 8590833907

Offset: 0

Author

Gus Wiseman, Mar 02 2020

Keywords

• nonn

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.

Examples

			The a(5) = 1 through a(7) = 14 compositions:
  (212)  (213)   (1213)
         (312)   (1312)
         (1212)  (2113)
         (2112)  (2122)
         (2121)  (2131)
                 (2212)
                 (3112)
                 (3121)
                 (11212)
                 (12112)
                 (12121)
                 (21112)
                 (21121)
                 (21211)
		

Links

• MathWorld, Unimodal Sequence

Crossrefs

Not requiring non-unimodality gives A107429.

Not requiring the covering condition gives A115981.

The complement is counted by A227038.

A version for partitions is A332579, with complement A332577.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal normal sequences are A328509.

Numbers whose unsorted prime signature is not unimodal are A332282.

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

Programs

Mathematica

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&!unimodQ[#]&]],{n,0,10}]

Formula

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