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%I #5 Nov 15 2022 10:12:45
%S 1,2,3,4,5,7,8,9,11,13,15,16,17,21,25,27,29,31,32,37,41,43,49,53,57,
%T 59,61,63,64,65,73,81,85,101,105,107,113,117,121,123,125,127,128,129,
%U 137,145,165,169,171,193,201,209,213,229,233,235,241,245,249,251
%N Numbers k such that the k-th standard ordered rooted tree is fully canonically ordered (counted by A000081).
%C The ordering of finitary multisets is first by length and then lexicographically. This is also the ordering used for Mathematica expressions.
%C We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
%H 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 The terms together with their corresponding ordered rooted trees begin:
%e 1: o
%e 2: (o)
%e 3: ((o))
%e 4: (oo)
%e 5: (((o)))
%e 7: (o(o))
%e 8: (ooo)
%e 9: ((oo))
%e 11: ((o)(o))
%e 13: (o((o)))
%e 15: (oo(o))
%e 16: (oooo)
%e 17: ((((o))))
%e 21: ((o)((o)))
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t Select[Range[1000],FreeQ[srt[#],_[__]?(!OrderedQ[#]&)]&]
%Y These trees are counted by A000081.
%Y A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y Cf. A001263, A004249, A005043, A032027, A063895, A276625, A358373-A358377.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 14 2022
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