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%I #19 Jan 21 2024 11:07:14
%S 0,0,0,0,0,0,3,8,27,75,185,441,1025,2276,4985,10753,22863,48142,
%T 100583,208663,430563,884407,1809546,3690632,7506774,15233198,
%U 30851271,62377004,125934437,253936064,511491634,1029318958,2069728850,4158873540,8351730223,16762945432
%N Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
%C A composition of n is a finite sequence of positive integers summing to n.
%H Andrew Howroyd, <a href="/A332833/b332833.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).
%e The a(6) = 3 and a(7) = 8 compositions:
%e (1221) (2113)
%e (2112) (3112)
%e (11211) (11311)
%e (12112)
%e (21112)
%e (21121)
%e (111211)
%e (112111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,10}]
%Y The case of partitions is A332641.
%Y The version for unsorted prime signature is A332831.
%Y The version for the compositions themselves (not run-lengths) is A332834.
%Y The complement is counted by A332835.
%Y Unimodal compositions are A001523.
%Y Partitions with weakly increasing run-lengths are A100883.
%Y Compositions that are not unimodal are A115981.
%Y Compositions with equal run-lengths are A329738.
%Y Compositions whose run-lengths are unimodal are A332726.
%Y Compositions whose run-lengths are not unimodal are A332727.
%Y Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
%Y Compositions with weakly increasing run-lengths are A332836.
%Y Compositions that are neither unimodal nor is their negation are A332870.
%Y Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.
%K nonn
%O 0,7
%A _Gus Wiseman_, Feb 29 2020
%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020
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