A332833 - OEIS

%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