%I #6 Nov 14 2022 16:06:46
%S 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 21,22,23,24,26,27,28,29,30,31,32,34,35,36,41,49,50,57,66,97,130,193,
%U 258,385,513,514,769,1025,2049,4097,8193,16385,32769,65537,131073
%N 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 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 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 Triangle begins:
%e 1
%e 2
%e 3 4
%e 5 6 7 8 9
%e 10 11 12 13 14 15 16 17 18 25 33 65 129 257
%e 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 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t aotrank[t_]:=If[t=={},1,1+stcinv[aotrank/@t]];
%t aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
%t Table[Sort[aotrank/@aot[n]],{n,6}]
%Y The version for compositions is A000027.
%Y Row-lengths are A000108.
%Y The unordered version (using Matula-Goebel numbers) is A061773.
%Y The version for Heinz numbers of partitions is A215366.
%Y The row containing n is A358372(n).
%Y A000081 counts unlabeled rooted trees, ranked by A358378.
%Y A001263 counts unlabeled ordered rooted trees by nodes and leaves.
%Y A358371 counts leaves in standard ordered rooted trees.
%Y Cf. A004249, A005043, A061775, A109129, A284778, A358376, A358377.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Nov 14 2022