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A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

%I A332833 #20 Feb 16 2025 08:33:59
%S A332833 0,0,0,0,0,0,3,8,27,75,185,441,1025,2276,4985,10753,22863,48142,
%T A332833 100583,208663,430563,884407,1809546,3690632,7506774,15233198,
%U A332833 30851271,62377004,125934437,253936064,511491634,1029318958,2069728850,4158873540,8351730223,16762945432
%N A332833 Number of compositions of n whose run-lengths are neither weakly increasing nor weakly decreasing.
%C A332833 A composition of n is a finite sequence of positive integers summing to n.
%H A332833 Andrew Howroyd, <a href="/A332833/b332833.txt">Table of n, a(n) for n = 0..1000</a>
%H A332833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.
%F A332833 a(n) = 2^(n - 1) - 2 * A332836(n) + A329738(n).
%e A332833 The a(6) = 3 and a(7) = 8 compositions:
%e A332833   (1221)   (2113)
%e A332833   (2112)   (3112)
%e A332833   (11211)  (11311)
%e A332833            (12112)
%e A332833            (21112)
%e A332833            (21121)
%e A332833            (111211)
%e A332833            (112111)
%t A332833 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,10}]
%Y A332833 The case of partitions is A332641.
%Y A332833 The version for unsorted prime signature is A332831.
%Y A332833 The version for the compositions themselves (not run-lengths) is A332834.
%Y A332833 The complement is counted by A332835.
%Y A332833 Unimodal compositions are A001523.
%Y A332833 Partitions with weakly increasing run-lengths are A100883.
%Y A332833 Compositions that are not unimodal are A115981.
%Y A332833 Compositions with equal run-lengths are A329738.
%Y A332833 Compositions whose run-lengths are unimodal are A332726.
%Y A332833 Compositions whose run-lengths are not unimodal are A332727.
%Y A332833 Partitions with weakly increasing or weakly decreasing run-lengths: A332745.
%Y A332833 Compositions with weakly increasing run-lengths are A332836.
%Y A332833 Compositions that are neither unimodal nor is their negation are A332870.
%Y A332833 Cf. A001462, A072704, A072706, A107429, A181819, A329398, A329744, A329746, A329766, A332273, A332640, A332746.
%K A332833 nonn
%O A332833 0,7
%A A332833 _Gus Wiseman_, Feb 29 2020
%E A332833 Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020