OFFSET
1,2
COMMENTS
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.
LINKS
EXAMPLE
Triangle begins:
1
2
3 4
5 6 7 8 9
10 11 12 13 14 15 16 17 18 25 33 65 129 257
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.
MATHEMATICA
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
aotrank[t_]:=If[t=={}, 1, 1+stcinv[aotrank/@t]];
aot[n_]:=If[n==1, {{}}, Join@@Table[Tuples[aot/@c], {c, Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Sort[aotrank/@aot[n]], {n, 6}]
CROSSREFS
The version for compositions is A000027.
Row-lengths are A000108.
The unordered version (using Matula-Goebel numbers) is A061773.
The version for Heinz numbers of partitions is A215366.
The row containing n is A358372(n).
A001263 counts unlabeled ordered rooted trees by nodes and leaves.
A358371 counts leaves in standard ordered rooted trees.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Nov 14 2022
STATUS
approved