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%I #8 Feb 26 2020 17:19:16
%S 0,0,0,0,0,0,0,0,1,1,3,5,9,14,22,33,48,69,96,136,184,248,330,443,574,
%T 756,970,1252,1595,2040,2558,3236,4041,5054,6256,7781,9547,11782,
%U 14394,17614,21423,26083,31501,38158,45930,55299,66262,79477,94803,113214
%N Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
%C Also partitions whose run-lengths and negated run-lengths are not both unimodal. 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(8) = 1 through a(13) = 14 partitions:
%e (3221) (4221) (5221) (4331) (4332) (5332)
%e (32221) (6221) (5331) (6331)
%e (33211) (42221) (7221) (8221)
%e (322211) (43221) (43321)
%e (332111) (44211) (44311)
%e (52221) (53221)
%e (322221) (62221)
%e (422211) (332221)
%e (3321111) (333211)
%e (422221)
%e (442111)
%e (522211)
%e (3222211)
%e (33211111)
%t Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]
%Y The complement is counted by A332745.
%Y The Heinz numbers of these partitions are A332831.
%Y The case of run-lengths of compositions is A332833.
%Y Partitions whose run-lengths are weakly increasing are A100883.
%Y Partitions whose run-lengths are weakly decreasing are A100882.
%Y Partitions whose run-lengths are not unimodal are A332281.
%Y Partitions whose negated run-lengths are not unimodal are A332639.
%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 Compositions whose negation is not unimodal are A332669.
%Y The case of run-lengths of compositions is A332833.
%Y Compositions that are neither increasing nor decreasing are A332834.
%Y Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.
%K nonn
%O 0,11
%A _Gus Wiseman_, Feb 26 2020