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A358330 By concatenating the standard compositions of each part of the a(n)-th standard composition, we get a weakly increasing sequence.

%I A358330 #10 Nov 11 2022 08:08:49
%S A358330 0,1,2,3,4,6,7,8,9,10,12,14,15,18,19,24,25,26,28,30,31,32,36,38,39,40,
%T A358330 42,50,51,56,57,58,60,62,63,64,72,73,74,76,78,79,96,100,102,103,104,
%U A358330 106,114,115,120,121,122,124,126,127,128,129,130,136,146,147
%N A358330 By concatenating the standard compositions of each part of the a(n)-th standard composition, we get a weakly increasing sequence.
%C A358330 Note we shorten the language, "the k-th composition in standard order," to "the standard composition of k."
%C A358330 The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.
%H A358330 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A358330 The terms together with their standard compositions begin:
%e A358330    0: ()
%e A358330    1: (1)
%e A358330    2: (2)
%e A358330    3: (1,1)
%e A358330    4: (3)
%e A358330    6: (1,2)
%e A358330    7: (1,1,1)
%e A358330    8: (4)
%e A358330    9: (3,1)
%e A358330   10: (2,2)
%e A358330   12: (1,3)
%e A358330   14: (1,1,2)
%e A358330   15: (1,1,1,1)
%e A358330   18: (3,2)
%e A358330   19: (3,1,1)
%e A358330   24: (1,4)
%e A358330   25: (1,3,1)
%e A358330   26: (1,2,2)
%e A358330 For example, the 532,488-th composition is (6,10,4), with standard compositions ((1,2),(2,2),(3)), with weakly increasing concatenation (1,2,2,2,3), so 532,488 is in the sequence.
%t A358330 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358330 Select[Range[0,100],OrderedQ[Join@@stc/@stc[#]]&]
%Y A358330 See link for sequences related to standard compositions.
%Y A358330 Standard compositions are listed by A066099.
%Y A358330 Indices of rows of A357135 (ranked by A357134) that are weakly increasing.
%Y A358330 Cf. A000120, A001511, A029931, A048896, A058891, A070939, A333766, A335404, A357137, A357186.
%K A358330 nonn
%O A358330 1,3
%A A358330 _Gus Wiseman_, Nov 10 2022