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%I #15 Dec 31 2020 19:17:10
%S 0,0,0,0,1,4,8,24,55,128,282,625,1336,2855,6000,12551,26022,53744,
%T 110361,225914,460756,937413,1902370,3853445,7791647,15732468,
%U 31725191,63907437,128613224,258626480,519700800,1043690354,2094882574,4202903667,8428794336,16897836060
%N Number of compositions of n whose run-lengths are not weakly increasing.
%C A composition of n is a finite sequence of positive integers summing to n.
%C Also compositions whose run-lengths are not weakly decreasing.
%H Andrew Howroyd, <a href="/A332871/b332871.txt">Table of n, a(n) for n = 0..1000</a>
%F a(n) = 2^(n - 1) - A332836(n).
%e The a(4) = 1 through a(6) = 8 compositions:
%e (112) (113) (114)
%e (221) (1113)
%e (1112) (1131)
%e (1121) (1221)
%e (2112)
%e (11112)
%e (11121)
%e (11211)
%e For example, the composition (2,1,1,2) has run-lengths (1,2,1), which are not weakly increasing, so (2,1,1,2) is counted under a(6).
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!LessEqual@@Length/@Split[#]&]],{n,0,10}]
%Y The version for the compositions themselves (not run-lengths) is A056823.
%Y The version for unsorted prime signature is A112769, with dual A071365.
%Y The case without weakly decreasing run-lengths either is A332833.
%Y The complement is counted by A332836.
%Y Compositions that are not unimodal are A115981.
%Y Compositions with equal run-lengths are A329738.
%Y Compositions whose run-lengths are not unimodal are A332727.
%Y Cf. A001523, A072704, A100883, A181819, A329744, A329766, A332641, A332669, A332726, A332745, A332746, A332834, A332835.
%K nonn
%O 0,6
%A _Gus Wiseman_, Feb 29 2020
%E Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020