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%I #5 May 18 2020 06:37:50
%S 0,0,0,0,0,0,4,4,8,12,38,42,72,98,150,298,372,542,760,1070,1428,2600,
%T 3120,4550,6050,8478,10976,15220,23872,29950,41276,55062,74096,97148,
%U 129786,167256,256070,314454,429338,556364,749266,955746,1275016,1618054
%N Number of strict compositions of n that are neither increasing nor decreasing.
%C A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%F a(n) = A032020(n) - 2*A000009(n) + 1.
%e The a(6) = 4 through a(9) = 12 compositions:
%e (1,3,2) (1,4,2) (1,4,3) (1,5,3)
%e (2,1,3) (2,1,4) (1,5,2) (1,6,2)
%e (2,3,1) (2,4,1) (2,1,5) (2,1,6)
%e (3,1,2) (4,1,2) (2,5,1) (2,4,3)
%e (3,1,4) (2,6,1)
%e (3,4,1) (3,1,5)
%e (4,1,3) (3,2,4)
%e (5,1,2) (3,4,2)
%e (3,5,1)
%e (4,2,3)
%e (5,1,3)
%e (6,1,2)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!Greater@@#&&!Less@@#&]],{n,0,10}]
%Y The non-strict case is A332834.
%Y The complement is counted by A333147.
%Y Strict partitions are A000009.
%Y Strict compositions are A032020.
%Y Non-unimodal strict compositions are A072707.
%Y Strict partitions with increasing or decreasing run-lengths are A333190.
%Y Strict compositions with increasing or decreasing run-lengths are A333191.
%Y Unimodal compositions are A001523, with strict case A072706.
%Y Cf. A059204, A115981, A227038, A329398, A332745, A332746, A332831, A332833, A332835, A332874, A333150, A333192.
%K nonn
%O 0,7
%A _Gus Wiseman_, May 16 2020
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