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%I #14 Jan 21 2024 11:07:20
%S 0,0,0,0,1,4,14,36,88,199,432,914,1900,3896,7926,16036,32311,64944,
%T 130308,261166,523040,1046996,2095152,4191796,8385466,16773303,
%U 33549564,67102848,134210298,268426328,536859712,1073728142,2147466956,4294947014,8589909976
%N Number of compositions of n that 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="/A332834/b332834.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 * A000041(n) + A000005(n).
%e The a(4) = 1 through a(6) = 14 compositions:
%e (121) (131) (132)
%e (212) (141)
%e (1121) (213)
%e (1211) (231)
%e (312)
%e (1131)
%e (1212)
%e (1221)
%e (1311)
%e (2112)
%e (2121)
%e (11121)
%e (11211)
%e (12111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Or[LessEqual@@#,GreaterEqual@@#]&]],{n,0,10}]
%o (PARI) a(n)={if(n==0, 0, 2^(n-1) - 2*numbpart(n) + numdiv(n))} \\ _Andrew Howroyd_, Dec 30 2020
%Y The version for unsorted prime signature is A332831.
%Y The version for run-lengths of compositions is A332833.
%Y The complement appears to be counted by A329398.
%Y Unimodal compositions are A001523.
%Y Compositions that are not unimodal are A115981.
%Y Partitions with weakly increasing or decreasing run-lengths are A332745.
%Y Compositions with weakly increasing or decreasing run-lengths are A332835.
%Y Compositions with weakly increasing run-lengths are A332836.
%Y Compositions that are neither unimodal nor is their negation are A332870.
%Y Cf. A007052, A072704, A107429, A328509, A329744, A332281, A332284, A332578, A332640, A332641, A332643, A332669, A332746.
%K nonn
%O 0,6
%A _Gus Wiseman_, Feb 29 2020
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