%I #7 Oct 13 2020 14:34:35
%S 0,0,1,2,6,13,28,59,122,248,502,1012,2033,4078,8170,16357,32736,65498,
%T 131026,262090,524224,1048500,2097063,4194200,8388486,16777074,
%U 33554267,67108672,134217506,268435200,536870616,1073741484,2147483258,4294966848,8589934080
%N Number of compositions of n that are not strictly increasing.
%F a(n) = 2^(n-1) - A000009(n) for n > 0.
%e The a(2) = 1 through a(5) = 13 compositions:
%e (11) (21) (22) (32)
%e (111) (31) (41)
%e (112) (113)
%e (121) (122)
%e (211) (131)
%e (1111) (212)
%e (221)
%e (311)
%e (1112)
%e (1121)
%e (1211)
%e (2111)
%e (11111)
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&]],{n,0,15}]
%Y A000009 counts the complement.
%Y A047967 is the unordered version.
%Y A056823 is the weak version.
%Y A140106 counts the unordered case of length 3.
%Y A242771 counts the case of length 3.
%Y A333255 is the complement of a ranking sequence (using standard compositions A066099) for these compositions.
%Y A337481 counts these compositions that are not strictly decreasing.
%Y A337482 counts these compositions that are not weakly decreasing.
%Y A001523 counts unimodal compositions, with complement A115981.
%Y A007318 and A097805 count compositions by length.
%Y A218004 counts strictly increasing or weakly decreasing compositions.
%Y Cf. A000212, A128422, A332745, A332834, A332835, A337484.
%K nonn
%O 0,4
%A _Gus Wiseman_, Oct 06 2020