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Numbers k such that the k-th composition in standard order is strictly increasing.
46

%I #4 Mar 21 2020 16:35:30

%S 0,1,2,4,6,8,12,16,20,24,32,40,48,52,64,72,80,96,104,128,144,160,192,

%T 200,208,256,272,288,320,328,384,400,416,512,544,576,640,656,768,784,

%U 800,832,840,1024,1056,1088,1152,1280,1296,1312,1536,1568,1600,1664,1680

%N Numbers k such that the k-th composition in standard order is strictly increasing.

%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) 656: (2,3,5)

%e 2: (2) 144: (3,5) 768: (1,9)

%e 4: (3) 160: (2,6) 784: (1,4,5)

%e 6: (1,2) 192: (1,7) 800: (1,3,6)

%e 8: (4) 200: (1,3,4) 832: (1,2,7)

%e 12: (1,3) 208: (1,2,5) 840: (1,2,3,4)

%e 16: (5) 256: (9) 1024: (11)

%e 20: (2,3) 272: (4,5) 1056: (5,6)

%e 24: (1,4) 288: (3,6) 1088: (4,7)

%e 32: (6) 320: (2,7) 1152: (3,8)

%e 40: (2,4) 328: (2,3,4) 1280: (2,9)

%e 48: (1,5) 384: (1,8) 1296: (2,4,5)

%e 52: (1,2,3) 400: (1,3,5) 1312: (2,3,6)

%e 64: (7) 416: (1,2,6) 1536: (1,10)

%e 72: (3,4) 512: (10) 1568: (1,4,6)

%e 80: (2,5) 544: (4,6) 1600: (1,3,7)

%e 96: (1,6) 576: (3,7) 1664: (1,2,8)

%e 104: (1,2,4) 640: (2,8) 1680: (1,2,3,5)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,1000],Less@@stc[#]&]

%Y Strictly increasing runs are counted by A124768.

%Y The normal case is A164894.

%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 decreasing version is A333256.

%Y Cf. A000120, A029931, A048793, A066099, A070939, A124762, A228351, A333217, A333220, A333379.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 20 2020