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%I #4 Mar 21 2020 16:35:38
%S 0,1,2,4,5,8,9,16,17,18,32,33,34,37,64,65,66,68,69,128,129,130,132,
%T 133,137,256,257,258,260,261,264,265,274,512,513,514,516,517,520,521,
%U 529,530,549,1024,1025,1026,1028,1029,1032,1033,1040,1041,1042,1058,1061
%N Numbers k such that the k-th composition in standard order is strictly decreasing.
%C 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 The sequence of positive terms together with the corresponding compositions begins:
%e 1: (1) 128: (8) 517: (7,2,1)
%e 2: (2) 129: (7,1) 520: (6,4)
%e 4: (3) 130: (6,2) 521: (6,3,1)
%e 5: (2,1) 132: (5,3) 529: (5,4,1)
%e 8: (4) 133: (5,2,1) 530: (5,3,2)
%e 9: (3,1) 137: (4,3,1) 549: (4,3,2,1)
%e 16: (5) 256: (9) 1024: (11)
%e 17: (4,1) 257: (8,1) 1025: (10,1)
%e 18: (3,2) 258: (7,2) 1026: (9,2)
%e 32: (6) 260: (6,3) 1028: (8,3)
%e 33: (5,1) 261: (6,2,1) 1029: (8,2,1)
%e 34: (4,2) 264: (5,4) 1032: (7,4)
%e 37: (3,2,1) 265: (5,3,1) 1033: (7,3,1)
%e 64: (7) 274: (4,3,2) 1040: (6,5)
%e 65: (6,1) 512: (10) 1041: (6,4,1)
%e 66: (5,2) 513: (9,1) 1042: (6,3,2)
%e 68: (4,3) 514: (8,2) 1058: (5,4,2)
%e 69: (4,2,1) 516: (7,3) 1061: (5,3,2,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[0,1000],Greater@@stc[#]&]
%Y Strictly increasing runs are counted by A124768.
%Y The normal case is A246534.
%Y The weakly decreasing version is A114994.
%Y The weakly increasing version is A225620.
%Y The unequal version is A233564.
%Y The equal version is A272919.
%Y The strictly increasing version is A333255.
%Y Cf. A000120, A029931, A048793, A066099, A070939, A124769, A228351, A233249, A333217, A333256, A333380.
%K nonn
%O 1,3
%A _Gus Wiseman_, Mar 20 2020
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