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A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.

%I A358373 #6 Nov 14 2022 16:06:46
%S A358373 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,25,33,65,129,257,19,20,
%T A358373 21,22,23,24,26,27,28,29,30,31,32,34,35,36,41,49,50,57,66,97,130,193,
%U A358373 258,385,513,514,769,1025,2049,4097,8193,16385,32769,65537,131073
%N A358373 Triangle read by rows where row n lists the sorted standard ordered rooted tree-numbers of all unlabeled ordered rooted trees with n vertices.
%C A358373 We define the standard ordered rooted tree (SORT)-number of an unlabeled ordered rooted tree to be one plus the standard composition number (A066099) of the SORT-numbers of the branches, or 1 if there are no branches. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
%H A358373 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 A358373 Triangle begins:
%e A358373    1
%e A358373    2
%e A358373    3   4
%e A358373    5   6   7   8   9
%e A358373   10  11  12  13  14  15  16  17  18  25  33  65 129 257
%e A358373 For example, the tree t = ((o,o),o) has branches (o,o) and o with SORT-numbers 4 and 1, and the standard composition number of (4,1) is 17, so t has SORT-number 18 and is found in row 5.
%t A358373 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A358373 aotrank[t_]:=If[t=={},1,1+stcinv[aotrank/@t]];
%t A358373 aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t A358373 Table[Sort[aotrank/@aot[n]],{n,6}]
%Y A358373 The version for compositions is A000027.
%Y A358373 Row-lengths are A000108.
%Y A358373 The unordered version (using Matula-Goebel numbers) is A061773.
%Y A358373 The version for Heinz numbers of partitions is A215366.
%Y A358373 The row containing n is A358372(n).
%Y A358373 A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A358373 A001263 counts unlabeled ordered rooted trees by nodes and leaves.
%Y A358373 A358371 counts leaves in standard ordered rooted trees.
%Y A358373 Cf. A004249, A005043, A061775, A109129, A284778, A358376, A358377.
%K A358373 nonn,tabf
%O A358373 1,2
%A A358373 _Gus Wiseman_, Nov 14 2022