A332640 Number of integer partitions of n such that neither the run-lengths nor the negated run-lengths are unimodal.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 17, 29, 44, 66, 92, 138, 187, 266, 359, 492, 649, 877, 1140, 1503, 1938, 2517, 3202, 4111, 5175, 6563, 8209, 10297, 12763, 15898, 19568, 24152, 29575, 36249, 44090, 53737, 65022, 78752, 94873, 114294

Offset: 0

Author

Gus Wiseman, Feb 25 2020

Keywords

• nonn

Comments

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

Examples

			The a(14) = 1 through a(18) = 12 partitions:
  (433211)  (533211)   (443221)    (544211)     (544311)
            (4332111)  (633211)    (733211)     (553221)
                       (5332111)   (4333211)    (644211)
                       (43321111)  (6332111)    (833211)
                                   (53321111)   (4432221)
                                   (433211111)  (5333211)
                                                (5442111)
                                                (7332111)
                                                (43332111)
                                                (63321111)
                                                (533211111)
                                                (4332111111)
For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).
		

Links

• MathWorld, Unimodal Sequence

Crossrefs

Looking only at the original run-lengths gives A332281.

Looking only at the negated run-lengths gives A332639.

The Heinz numbers of these partitions are A332643.

The complement is counted by A332746.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Partitions with unimodal run-lengths are A332280.

Partitions whose negated run-lengths are unimodal are A332638.

Run-lengths and negated run-lengths are not both unimodal: A332641.

Compositions whose negation is not unimodal are A332669.

Run-lengths and negated run-lengths are both unimodal: A332745.

Cf. A007052, A025065, A100883, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332642, A332726, A332727.

Programs

Mathematica

unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n],!unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]],{n,0,30}]