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A358374 Numbers k such that the k-th standard ordered rooted tree is an identity tree (counted by A032027).

%I A358374 #5 Nov 14 2022 20:01:14
%S A358374 1,2,3,5,6,7,10,13,17,19,21,33,34,38,39,42,45,49,51,53,65,66,67,81,97,
%T A358374 130,131,133,134,135,145,161,162,177,193,195,209,259,261,262,263,266,
%U A358374 269,289,290,305,321,322,353,387,389,401,417,513,517,518,519,522
%N A358374 Numbers k such that the k-th standard ordered rooted tree is an identity tree (counted by A032027).
%C A358374 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.
%C A358374 A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root.
%H A358374 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 A358374 The terms together with their corresponding ordered rooted trees begin:
%e A358374    1: o
%e A358374    2: (o)
%e A358374    3: ((o))
%e A358374    5: (((o)))
%e A358374    6: ((o)o)
%e A358374    7: (o(o))
%e A358374   10: (((o))o)
%e A358374   13: (o((o)))
%e A358374   17: ((((o))))
%e A358374   19: (((o))(o))
%e A358374   21: ((o)((o)))
%e A358374   33: (((o)o))
%e A358374   34: ((((o)))o)
%e A358374   38: (((o))(o)o)
%e A358374   39: (((o))o(o))
%e A358374   42: ((o)((o))o)
%e A358374   45: ((o)o((o)))
%t A358374 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A358374 srt[n_]:=If[n==1,{},srt/@stc[n-1]];
%t A358374 Select[Range[100],FreeQ[srt[#],_[__]?(!UnsameQ@@#&)]&]
%Y A358374 These trees are counted by A032027.
%Y A358374 The unordered version is A276625, counted by A004111.
%Y A358374 A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358374 A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
%Y A358374 A358375 ranks ordered binary trees, counted by A126120.
%Y A358374 Cf. A001263, A004249, A005043, A063895, A358373, A358376, A358377.
%K A358374 nonn
%O A358374 1,2
%A A358374 _Gus Wiseman_, Nov 14 2022