A332640 - OEIS

%I #9 Feb 26 2020 17:19:07

%S 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,

%T 359,492,649,877,1140,1503,1938,2517,3202,4111,5175,6563,8209,10297,

%U 12763,15898,19568,24152,29575,36249,44090,53737,65022,78752,94873,114294

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

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

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

%e The a(14) = 1 through a(18) = 12 partitions:

%e   (433211)  (533211)   (443221)    (544211)     (544311)

%e             (4332111)  (633211)    (733211)     (553221)

%e                        (5332111)   (4333211)    (644211)

%e                        (43321111)  (6332111)    (833211)

%e                                    (53321111)   (4432221)

%e                                    (433211111)  (5333211)

%e                                                 (5442111)

%e                                                 (7332111)

%e                                                 (43332111)

%e                                                 (63321111)

%e                                                 (533211111)

%e                                                 (4332111111)

%e For example, the partition (4,3,3,2,1,1) has run-lengths (1,2,1,2), so is counted under a(14).

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

%t Table[Length[Select[IntegerPartitions[n],!unimodQ[Length/@Split[#]]&&!unimodQ[-Length/@Split[#]]&]],{n,0,30}]

%Y Looking only at the original run-lengths gives A332281.

%Y Looking only at the negated run-lengths gives A332639.

%Y The Heinz numbers of these partitions are A332643.

%Y The complement is counted by A332746.

%Y Unimodal compositions are A001523.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Partitions with unimodal run-lengths are A332280.

%Y Partitions whose negated run-lengths are unimodal are A332638.

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

%Y Compositions whose negation is not unimodal are A332669.

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

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

%K nonn

%O 0,16

%A _Gus Wiseman_, Feb 25 2020