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A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.

%I A333256 #4 Mar 21 2020 16:35:38
%S A333256 0,1,2,4,5,8,9,16,17,18,32,33,34,37,64,65,66,68,69,128,129,130,132,
%T A333256 133,137,256,257,258,260,261,264,265,274,512,513,514,516,517,520,521,
%U A333256 529,530,549,1024,1025,1026,1028,1029,1032,1033,1040,1041,1042,1058,1061
%N A333256 Numbers k such that the k-th composition in standard order is strictly decreasing.
%C A333256 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again.
%e A333256 The sequence of positive terms together with the corresponding compositions begins:
%e A333256      1: (1)         128: (8)         517: (7,2,1)
%e A333256      2: (2)         129: (7,1)       520: (6,4)
%e A333256      4: (3)         130: (6,2)       521: (6,3,1)
%e A333256      5: (2,1)       132: (5,3)       529: (5,4,1)
%e A333256      8: (4)         133: (5,2,1)     530: (5,3,2)
%e A333256      9: (3,1)       137: (4,3,1)     549: (4,3,2,1)
%e A333256     16: (5)         256: (9)        1024: (11)
%e A333256     17: (4,1)       257: (8,1)      1025: (10,1)
%e A333256     18: (3,2)       258: (7,2)      1026: (9,2)
%e A333256     32: (6)         260: (6,3)      1028: (8,3)
%e A333256     33: (5,1)       261: (6,2,1)    1029: (8,2,1)
%e A333256     34: (4,2)       264: (5,4)      1032: (7,4)
%e A333256     37: (3,2,1)     265: (5,3,1)    1033: (7,3,1)
%e A333256     64: (7)         274: (4,3,2)    1040: (6,5)
%e A333256     65: (6,1)       512: (10)       1041: (6,4,1)
%e A333256     66: (5,2)       513: (9,1)      1042: (6,3,2)
%e A333256     68: (4,3)       514: (8,2)      1058: (5,4,2)
%e A333256     69: (4,2,1)     516: (7,3)      1061: (5,3,2,1)
%t A333256 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333256 Select[Range[0,1000],Greater@@stc[#]&]
%Y A333256 Strictly increasing runs are counted by A124768.
%Y A333256 The normal case is A246534.
%Y A333256 The weakly decreasing version is A114994.
%Y A333256 The weakly increasing version is A225620.
%Y A333256 The unequal version is A233564.
%Y A333256 The equal version is A272919.
%Y A333256 The strictly increasing version is A333255.
%Y A333256 Cf. A000120, A029931, A048793, A066099, A070939, A124769, A228351, A233249, A333217, A333256, A333380.
%K A333256 nonn
%O A333256 1,3
%A A333256 _Gus Wiseman_, Mar 20 2020