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

0,7

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

Table of n, a(n) for n=0..35.

MathWorld, Unimodal Sequence

Formula

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

Example

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)

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}]

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.

Sequence in context: A027974 A027983 A142585 * A234097 A211562 A261055

Adjacent sequences: A332740 A332741 A332742 * A332744 A332745 A332746

Keyword

nonn

Author

Gus Wiseman, Mar 02 2020

Status

approved