A332834 - OEIS

%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